Well here we go. My custom Amumu build. Designed to go in, Hit for half the health of anyone around, and then run away and let others finish the job. I recognize that why it does have some tanky stuff, it doesn't excel at it, better able to act as a Hexplosive and temporary shield then anything.

Pros:

High damage
Not squishy.
Great in team fights.

Cons:

Not a tank
The burst damage isn't enough to kill off a person in one go.

The Defense tree will fill in the gaps of his defense that the item build creates. I would like to note that I myself am not level 30 so I have yet to full test around the Masteries but in theory that is the best line.

The choices of items was designed to increase his damage output as much as possible while still letting him take a few hits.

The first item should always be the fairy charm, because Amumu can be mana hungry and every little bit helps. The Philosopher's stone is really just to help accelerate the item build, but the regen is a welcome bonus as well, allowing Amumu to last longer before having to port back to base, The more farming the better after all.

The sunfire cloak, I have tired putting that later in the build and, Amumu just can't be without it in my opinion. It acts like a mini despair while also adding to his defense, again, the longer Amumu can be out there before having to retreat the better, and this gives him an offensive edge as well.

Next are the shoes. I put the mercury treads down because they are always a good item to have, but they can be replaced with any of the following: Ninja Tabi for added armour, Sorcerers shoes for a little extra punch, or Ionian Boots of Lucidity for fast cooldown, or even boots of mobility if you are jumping lanes to help out. Use your head and make the best choice for the current match.

The Arch Angels Staff: Some of you may say that this isn't a good choice for Amumu, and I almost agree with you if I didn't have the belief that it's great for any AP build. A Massive amount of extra mana, that grows as you use your abilities, The AP Bonus is the biggest within the game if you count in all the bonus from the mana. and the mana REgen it gives is also the best. While it's true he can't take full advantage of it, it's still a wonderful item.

The next three items, Abyssal Scepter, Frost Shield and Will of the Ancients are all choosen for One reason, They help your teammates as much as they help you. They can be purchased in any order depending on the match,

As stated before, I havn't reached level 30 yet, nor lvl 20. so I have yet to really play with the ruins. Armour, Ap, Mana Reg, Mana, and cooldown reduction are all valid choices.