What build grants the maximum amount of AP? What build grants the maximum amount of AD? What build grants max health? What build grants max armor? What build grants max MR?
What build grant the maximum efficiency of these?
Efficient as in no overlapping unique passives, having boots, and other stats not lacking as much as with a full AP/AD build, etc...
This does not include champions.
What build grants the maximum amount of AP?
Veigar with anything.
Otherwise, Rabadon's Deathcap, Seraph's Embrace, 3x Rod of Ages all stacked, Meijai's Soulstealer max stacked.
What build grants the maximum amount of AD?
Jhin with 5x 75 AD items and any upgraded Zeal Item.
What build grants max health?
Sion with anything.
Otherwise, Cinderhulk and 5x Warmog's Armor
What build grants max armor?
Rammus (maybe Malphite) with 6x Thornmail's.
What build grants max MR?
6x Spirit Visage or 6x Banshee's Veil
What build grant the maximum efficiency of these? Efficient as in no overlapping unique passives, having boots, and other stats not lacking as much as with a full AP/AD build, etc...
That really, really depends. For example, the armor one could be better with 2-3 armor + hp items and 1-2 mr items, but the amount of them, which ones to pick, and even if you get any changes from game to game constantly.
if you are looking champion specific builds, 
are the way to go for max ap,ad,armor and mr in order.
Efficency is going to be very game dependent (and also champ dependent), as the amount of MR/Armour/Magic pen/Armour pen involved, as well as base stats, is going to change the usefulness of each of those stats independently.
Simply getting the raw amount of various things isn't terribly exciting since in almost all cases you want a mix. (E.g. 6 rabadons is going to be max AP, but CDR and M pen are also important stats for mages.)
Max HP: 
= 3795 HP
Max AP: 
= 1051.65 AP
Max Armor: 
= 425 Armor
Max MR: 
= 315 MR
Max AD: 
= 430 AD
Blood Thirster + Infinity Edge(Upgraded by the mobile forge)
+ Trinity Force(Upgraded by the mobile forge)
+ Sanguine Blade
+ Ravenous Hydra
+ Black Cleaver(Upgraded by the mobile forge)
+ Press the attack + Overheal + Alaclarity + Cut Down + Electrocute + A Hunter(Any really) + Nasus(with 1000+ stacks)
