Which champion is the hardest to kill when at or close to full build? Why? How did you build him or her? I saw a cho'gath the other day that fully stacked his R and went full tank. He was pretty much unkillable
Sion with arbitrarily large amounts of health.
Alternatively, full build Maokai is a bitch to deal with if you rely on low cooldown abilities to deal damage.
Garen, Rammus, Alistar*
Garen's tankyness is uncounterable. Garen gets 20% increased Armor and MR for free, and a raw 30% damage reduction. Rammus gets a stupid amount of Armor and MR. This lets them build lots of tank stats, but also capitalize on huge health pool better than other champions. Both are also normal abilities, so they can be uber tanks often. I think Garen's raw 30% reduction is stronger than the tank stats Rammus gets because you can't build penetration to counter it.
Alistar is the tankiest for the duration of his ult if he builds a ton of HP. Which is an ultimate CD for tankiness and not his normal build. So ehhh.
Mundo is countered too hard by Ignite, no one else has anything too special.
Leona is also fantastic with her W.
Just taking damage or also fighting back?
Probably either Alistar, Poppy, or Maokai for the same reason. DR can't be mitigated.
I'm going with Poppy.
shen if fed or full build is really tanky. also full build poppy with her passive.
singed isnt really a full tank like the rest of these guys. singed builds consist of the rylia/liandry combo with roa or tear. id say singed is off tank. but yeah idk why ppl voted singed.
Volibear -- For a bruiser, he's stupendously annoying to kill. Shyvana -- Same as Volibear. Shen -- Ugh. Cho'Gath -- Too much HP. Alistar -- 70% Damage reduction. Maokai -- Bruiser with damage reduction. Gragas -- He gets Maokai level damage reduction, yet has a self heal. Galio -- He's like Maokai, except he can taunt you.
Singed is not a tank par say, he's more of an offtank that uses a frozen heart. He drops like a fly if you hit him with magic. Nautilus and Sejuani are not usually called up as unkillable as they lack sustain. Instead, they're walking CC machines.
i got a full build 
up to about 16.5k effective health from both sources 5.6k hp and over 200 of both ar and mr. his passive + spirit visage = free healing. ignite is an issue though. His size and lack of damage other than his w at that much health is also a problem and it's unlikely that you could get that build unless playing bots. Not sure if he's unkillable, but it'd be a pain to kill him without using all your cc on him to prevent him from healing
Bump
Nasus. He is unkillable with tank because he does mad damage and kills anyone that tries to kill him.
Trundle. His R destroys other tanks. Along with his passive and W he is a monster. He can also engage and disengage really fast with his W and E. All while stealing AD with his Q.
Well, whenever I have to fight a Malphite, it seems to take my entire team to actually deal damage to him.
no mundo
Rammus, you will kill yourself if you try using anything but abilities if he's already got a bit of armor.
