This build has been created to develop a beefy endgame Amumu so when commenting and voting keep that in mind. I am open to constructive criticism to help improve this build. I may make improvements in the future.

At first when I started developing my build I was using dodge runes and masteries as a starting point, but then read somewhere that stated your tantrum (E) ability's cooldown isn't reduced by .5 seconds when you dodge so I replaced those runes with armor since armor will help me out in the beginning with jungling, which seems to be the best choice with Amumu. You don't level as fast and someone must solo one of the side lanes by themselves for awhile, but you do earn a lot more money than you would if you were to lane.

When jungling in Summoner's Rift (5v5) start at the Blue Buff, which is a Golem, and work your way across the jungle using a Health Potion inbetween creeps or while fighting them. The Blue (mana) Buff regens your mana faster. If you don't start at the blue buff then you won't have enough mana to finish all the creeps on your first run or even half of them for that matter.

To start off the Golem have someone, preferably ranged, pull the creep for you by attacking it then running back to their lane before you attack it so that by the time it focuses on you you have already gotten a few shots in. Even for Amumu the golem is difficult to kill at level 1. I suggest using a health potion as soon as you initiate (begin) the battle with the Golem, otherwise you may die

I've also been ganked at the golem during my fight so to prevent from being ganked you may want the middle lane champion and the 1 or 2 champions, since you're jungling one of the side lanes are going to have to be played defensively by a single champion, from the nearest side lane to stay nearby until you have killed the golem. The farthest champion/s should just stay at their lane so that when the minions spawn they can be there right away, but if you think you may need the extra help by all means let them guard you. It's their loss in exp/gold.

I tend to use all 5 health potions by the time I have worked my way across my team's side of the jungle. You should have about 550G after your first run. I normally tele back, buy my Ninja Tabi (for the armor, not the dodge), buy 1 or 2 more health potions depending how much money I have left over, then go for a second run, which will only have 4 jungle creep camps, instead of the previous 6, this time since the buffs take longer to respawn. Depending how quickly you finish your second wave you should have some mana left and be at level 6, the level you can obtain your ultimate (R). By this time you may want to start ganking, but keep on jungling everytime you're near a creep camp and aren't needed right away since you need every bit of money you can get in the beginning. Just be smart about where you jungle and when.

Greater Glyph of Scaling Cooldown Reduction: x9 = 8.1% cooldown reduction
I picked this accumulative Glyph because the cooldowns matter more endgame than early game. I've had it happen many times before where 2 or 3 enemy champions barely escape after a big 5v5 team battle. (I'm also normally the last one to die) In those cases 1 or 2 extra Tantrums and a Bandage Toss (stun) could have finished them all off. Around the 25min mark an extra Bandage Toss or two could be very effective in earning you some kills in team ganks.

Greater Seal of Armor: x9 = 12.69 armor
These runes are meant to help the most in the beginning during you jungling stages, which is why I didn't pick the Greater Seal of Defense. Having the extra armor in the beginning will make it easier to survive while jungling the buffs and might allow you to use one less potion. I'm going to be buying most of my armor anyways, which is way better then a few petty runes.

Greater Mark of Magic Penetration: x9 = 8.55 magic penetration
80% of the damage you deal out as Amumu is magic damage so these runes seemed to be the best choice. Stack these runes with Amumu's passive (Amumu's attacks reduces the target's magic resistance for 4 seconds) and the Abyssal Scepter's (reduces nearby enemy targets' magic resistance by 20) and the enemy champion's magic resist may be brought down to 0 if they didn't buy any magic resistance.

Greater Quintessence of Armor: x3 = 12.78 armor
Additional armor for jungling.

I put most of my points in defense, obviously, since I'm building up a tank and the rest goes into the offense category. In the offense category I picked masteries that supported smite, which is necessary for jungling, AP, cooldown reduction, and magic penetration. Magic penetration is what makes Amumu so deadly. These stats are stacked by the runes as well.

I did not take Strength of Spirit since I will not have very much mana so as a result the health regen will be just as minuscule.

Of course, adjust some of the masteries according to the spells you take.

The first complete item I buy are Ninja Tabi. I would get them for the 25 armor, which can be useful to have right away and I don't mind dodging 10% of incoming basic attacks. My Tantrum cooldown won't be affected nearly enough for me to change this decision in choosing boots.

I do not get the items that provide extra gold per 5 sec because if you do the math those items do not earn you much gold in the end and don't offer very good stats compared to the Ninja Tabi.

I tend to get the Sunfire Aegis first so that I can solo the Dragon as soon as possible to get everyone gold. After I get the Ninja Tabi and Sunfire Cape I can solo the Dragon without health potions, but I'll be running low on mana after and have about 1/2 health with the possibly of being discovered and ganked. An immediate tele home should be my next course of action. If my team needs help ganking or the enemy team has some AP players I may want to hold off on the Sunfire Cape/Dragon and go with Force of Nature first.

If you buy the Sunfire Aegis first purchase the Chain Vest (armor) first so your jungling is easier, then Giant's Belt. A tank cannot survive with a lot of health and no defense or vice verse. Combine this with Amumu's Despair aura and you have a pretty deadly AOE aura.

If you buy Force of Nature first then you must be getting it for the magic resist so, obviously, buy the Negatron Cloak first. The extra speed helps you keep up with the enemy so that your aura can take effect and the health regen doesn't hurt either. The main reason I picked this item though is because it provides the most possible magic resist one item slot can give.

After buying both of those items I generally get Abyssal Mask for the additional magic resist and AP. The additional AP power is definitely enough to bring upon your enemies their demise. If your enemies are especially bad you can easily tank all 5 at once by yourself and kill 2 or 3 with the help of your ultimate.

The next item I buy is either another Force of Nature or another Abyssal Mask depending on which path I want to take.

Keep in mind the UNIQUE Passives doesn't stack with itself so the additional health regen won't take effect for the Force of Nature and neither does the magic penetration for the Abyssal Scepter, but I'm not getting the items for their passives. I'm getting them for their base stats. Look at the alternatives. Banshee's Veil is the next highest in magic resistance at 50. I don't need more mana or health, but the passive is fairly nice. It's a matter of opinion and preference.

After that, for my final item, I buy a Thornmail. A minimum of 15 damage must be received for the passive to take effect. That puts me over 300 armor, around 250 magic resist, and 70 - 140 AP. Your magic penetration stacks with basic attacks, runes, masteries, and items. Depending on the enemy line up I may skip the Thornmail all together and get another Abyssal Scepter.

Most the time I have a couple thousand gold left over at the end after I have finished my build depending how long the game lasts.

Escaping Tips - For when you're low on health
Make note of how much mana you have and use your abilities accordingly. If you are low then only use Despair and/or Bandage Toss and don't use Tantrum.

Cursed Touch: Can't really say this passive helps for escaping.

Bandage Toss: If there is only one champion chasing you use this combination: Bandage Toss, Despair, and Tantrum then run again. This will stun them and deal a considerable amount of damage, stalling your death for a second. Rinse and repeat and if you're lucky you may kill them before they kill you or a teammate might save you.

Despair: When you're being chased and the enemy is RIGHT on your tail turn Despair on so that they take damage as they chase you providing opportunity for a team gank on a weakened foe. A good team will notice when you're in trouble.

Tantrum: If you know your going to die then use this ability as much as possible and Despair to deliver as much damage as possible then keep running. Use this strategy for when there is more than one champion on your tail.

Curse of the Sad Mummy: Normally I don't suggest using this ability for escaping so that you'll have it for the nearby future, but if there are allies nearby, multiple enemy champions, and you have a chance to escape definitely use your ultimate, dealing a considerable amount of damage and stunning every enemy in your radius. That stun may buy you enough time to escape and your allies to clean up.

Skill Sequence
The only way to kill the Golem and Lizard at level one is by using Despair since it cuts right into their health by a percentage of their health as magic damage along with some additional magic damage.

My next choice would be Tantrum since its passive contributes to your armor based on its level and you can use it fairly often for a petty cost of 50 mana, which doesn't increase with level.

By level 3 you're still jungling so another Despair is helpful plus in mid to late game Despair is an easy way to take down tanks and/or multiple champions at once since it's an aura.

Again pick Tantrum to further increase your armor and cheap damage output.

After that pick Bandage Toss for its stun. After the first level of Bandage Toss I don't level any more until I have to since I only get it for the stun. If the stun duration were to increase with level I would upgrade it more, but that is not the case.

Then at level 6 your ultimate, Curse of the Sad Mummy, is available.

After this point I upgrade Despair and Curse of the Sad Mummy to the max ASAP. The ultimate always comes first because of obvious reasons. (Damage output and stun, the stun being more valuable in most cases) Once I have maxed Despair and are at levels you can't level your ulti I pick Tantrum for the damage output. Curse of the Sad Mummy is available at levels 6, 11, and 16. Lastly I upgrade Bandage Toss. Bandage Toss does do more damage to a single target than Tantrum, but Tantrum has an AOE so in team battles Tantrum is way more valuable, especially when demolishing built up mobs in lanes. Happy farming. ;)

Smite is necessary for this build, which leaves only one slot left.

Preferable Spells

Ghost: Great for getting around the jungle easily. Also great for chasing and escaping. Rather than attacking your enemies run along side them so your aura can take effect. It does more damage than your basic attacks.

Flash: Great for chasing enemies and escaping. When trying to escape in the jungle try flashing across walls to gain more distance. Some people can't tell which direction you flashed either, leaving them confused as to where you went.

Spells I'm Indifferent About

Heal: Most of your beefiness comes from your armor and magic resist, not your health.
Restoring 420+ health is more valuable the more defense you have so Heal could be valuable mid game and MAYBE end game, but I don't prefer this one since that means giving up flash or ghost. If you're low on health just return home.

Exhaust: This could be more valuable to your team rather than yourself. Remember, you have 2 stuns and one of them pulls you to the enemy, but you don't have very much damage output by yourself so if you were to stun the enemy then use exhaust your team could beat on him too.

I Would NEVER Use
Clairvoyance: Seeing an area of the map for 6 seconds every 2 minutes is not worth a slot to me.

Clarity: Just get the blue buff if you need mana. True that it does restore mana to your allies, but that's 50% less than what you would receive. Some champions don't even use mana. Tryndamere uses rage.

Rally: The amount of damage this gives you is so minuscule that it's hardly noticeable. Also half the champions use AP, not AD. Another point to cover is that it gives your minions AD as well, but my argument is why would you want the minions to be stronger? They would steal your minion kills, which equals money.

Teleport: You can only use teleport on a minion, turret, or pet. I don't see any of those in the jungle. Also Amumu's Skill set doesn't support teleport ganking well.

Cleanse: This spell is meant for people who have to run away often. Amumu is a tank and shouldn't be dying too often and his skill set kinda supports being chased if you use him right.

Fortify: If they reach your base you can easily defend better than a turret ever did.

Ignite: This spell is meant for finishing off champions that are running away while laning. You aren't laning. You're jungling.

Revive: You shouldn't be dying often so this spell isn't needed. If you play Amumu right you should be dealing out a lot of damage too so if you're dying it's your own fault for being stupid or weak (not getting your build up fast enough).

I only mentioned some of the obvious reasons so any arguments you may have I might have already thought of and just didn't think it mattered for Amumu or that it wasn't worth giving up a slot. Please do post your comments anyways