Both builds are definitly not viable for rankeds! Play them at own risk.

I'm not 100% sure if a i calculated the Health of Alistar corrently. I added all the flat bonus values (6007,76) and multiplied this with all percentages together (28%) and added the Base Health (2278):
(6007,76 * 128 / 100) + 2278. 

If there is a misstake, feel free to edit this answer but provide the calculation formula.

Alistar (without any skills involved)

the most health any champion can obtain is 9967,9328, being a level 18 Alistar with:

Base:


HP@18: 2278


Runepage:


Full HP@18: 536,76


Masteries:


Veteran's Scars: 36
Juggernaut: 3%


Items:


5*Warmog's Armor: 5000
1*Spirit of the Ancient Golem: 200 and + 25% Bonus Health
Elixir of Fortitude: 235


Sion (with skills involved)

unlimited health with Soul Furnace

Nunu (with skills involved)

the most health any champion can obtain is 13195.95, being a level 18 Nunu with:

All health runes and masteries:


9 Scaling Marks of Health (+87,48HP)
9 Scaling Glyphs of Health (+87,48HP)
9 Percent Seals of Health (+216HP)
3 Percent Quintessences of Health (+145,8HP)
3 points in Veteran's Scars mastery (+36HP)
1 point in Juggernaut mastery (+3%)
5 Warmog's Armor (+5000HP)
1 Spirit of the Ancient Golem (+200HP +25%)
An Elixir of Fortitude (+235HP)
Wild Growth cast by a Lulu with the maximum amount of ability power. (+???HP)
Consume cast on a Golem jungle creep (+10%)


Which champion can have the highest maximum health?
I've seen "determine" used for two completely different meanings:

Zeus' mood will determine tomorrow's weather. (determine = control, dictate)

The weather forecaster will determine tomorrow's weather. (determine = forecast, predict)

Are these meanings used equally often, or is one much more common than the other?
