Just to prove that AP Rammus is not crazy, I did some quick calculations (although you could just try it out in the game and check the damage graphs afterwards).

The base ability power from items in this build is 170 not including the Elixir of Brilliance... that's respectable but not huge. Due to bonuses from the masteries tree, that gets an additional 0.6 per level and an overall 4% increase. Total ability power is thus 188, although for some reason Mobafire does not include masteries in the total displayed.

Now in an ideal situation, you powerball into someone (probably 2 champs take the hit), then activate tremors and DBC while surrounded by the entire enemy team. Assuming you didn't have to retreat, you may chase them down with a powerball to finish them off, but let's disregard that final powerball damage since we are being generous by assuming you rammed two people initially when beginning the teamfight.

So the bonus magic damage from the your two Abilty Power items breaks down like this:

Rammus base Ability Power = 0
Ability Power from items = 170
After applying masteries = 188

Team Fight

Powerball
Targets = 2
Rate = 1.0 * AP
Bonus Damage = 376

Tremors
Targets = 5
Rate = 0.3 * AP per second
Duration = 8 seconds
Bonus Damage = 2256

So the total bonus magic damage from buying 170 AP is up to 2632 damage per teamfight. The magic penetration aura will also help your casters to hit harder as well. Remember this is bonus damage on top of any damage that the skills would already do.

Turret Push

Tremors
Targets = 1
Rate = 0.3 * AP per second
Duration = 8 seconds
Bonus Damage = 451

Remember this is bonus damage on top of any damage that the skill would already do.


Conclusion

To put it in other terms, the modest amount of AP in this build will be enough to increase Rammus's Powerball TOTAL damage by 63% and his Tremors TOTAL damage by 29%. Not a shocking amount maybe, but still quite useful.

The core items of the build which increase your Area of Effect damage are these two:

Abyssal Scepter, and Sunfire Cape

Whichever items you change or remove, do not remove those two. Also, there are very few situations where you do not want a Thornmail, so that is the third most important item.

Thornmail, ton of armor and brutal passive

About Boots:

If you are just crazy about maximizing damage or the enemy team is hopeless then Sorcerer's Shoes I suppose are a viable choice. Mercury's Treads however, are not needed, as you already have Cleanse and the Willpower mastery.

However, since this build relies on moving during the team fight to obtain the best position for maximum continual AoE damage, the only realistic choice of boots should be these babies:

Boots of Swiftness

Next we come to the less essential items.

Zhonya's Hourglass

While expensive and not essential, the hourglass has such great synergy with this build it would make little sense not to take it. It gives a ton of ability power which helps your Soaring Slam and Powerball damage. It also has armor which complements your Spiked Shell and Defensive Ball Curl damage. Finally it has an active that will be very useful in the late game when there is so much damage going around that the enemy team could nuke you within an inch of your life faster than an eyeblink.

Two seconds of invulnerability may not seem like much, but in a late game team fight it can make all the difference in the world. The icing on the cake is that your Tremors damage will keep hitting the enemy team while you are invulnerable.

Finally the most expendable item:
Randuin's Omen

Now you may be asking yourself, why would we want to slow the enemy's attack rate? Isn't Rammus doing more retaliation damage the faster the enemy attacks him? True, he is doing more retaliation damage, but he is not invincible you know. Eventually he will die and we don't want everything to happen so swifty that we don't have time to react or escape before his lifebar is completely wiped out. Remember there are other teammates doing damage and it is not all up to Rammus to finish the job!

Another thing about Randuin's Omen is that it has some nice HP regen via Warden's Mail, something the build would otherwise lack, and that keeps his lifebar completely full even as he is jungling or powerballing into wave after wave of enemy minions.

Something else I find is that Rammus doesn't farm very well in the early game because his attack is slow and he can be harassed by ranged spells. The Heart of Gold addresses his early game problems, and I hate to sell it later; that would kind of defeat the purpose of earning the extra gold in the first place. Since Randuin's Omen builds from Heart of Gold that is another consideration in its favor. Finally, the active ability can be used offensively or defensively which adds to Rammus's tactical options.

As the tank, you are the initiator. The general steps for initiating the team fight are as follows:

1. Powerball into enemy (preferably squishy DPS like Ashe)
2. Immediately taunt them with Puncturing Taunt
3. Immediately activate Defensive Ball Curl and Soaring Slam
4. Hit them, chase them, or just move around while letting the whole enemy team take massive AoE damage
5. Activate Zhonya's Hourglass for a couple extra seconds of free AoE damage and to remove any target locks on you (turrets included)
6. If health is low, retreat (using Cleanse and/or Ghost if needed) until you are far enough from the enemy to Powerball back to safety
7. If health is not low, use Powerball and/or Ghost to chase the enemy
8. For added escaping/chasing power, slow the whole enemy team by activating Randuin's Omen (this also helps your wounded teammates escape)
9. Repeat from Step 1

With this build, Rammus is very good at reaching unattended lanes swiftly, farming very easily, and then destroying turrets or masterfully escaping ganks. (However, you might not want to push too far if the enemy team has a lot of crowd control and your summoner spells are on cooldown.)

The general steps for farming during late game as quickly as possible are as follows:

1. Powerball ahead of your minion wave and ram into the enemy minion wave, trying to ram as many of them at once as possible
2. Activate Defensive Ball Curl
3. Save Soaring Slam for when you reach the enemy turret (unless it is very far away and/or you don't plan to reach it)
4. Stand directly in the middle of the enemy minion wave and keep attacking the ones with the lowest health until they are all dead
5. If there are any super minions, taunt them with Puncturing Taunt to maximize the initial damage

Rammus is also a decent jungler with this build. The Defensive Ball Curl and Soaring Slam ability make it pretty simple to demolish neutral minions. If you get a chance, pick up the Blue Golem's buff because it will give Rammus a great boost.

When you combine the Blue Golem buff with the 9 Greater Glyph of Scaling Cooldown Reduction runes, the passive of Randuin's Omen and the boost of Elixir of Brilliance, Rammus's cooldown reduction will be perfectly maxed out at just over 40%.