*THERE IS NO TL;DR. If it's TL, then DR and just hit BACK.*

N^2 example used is Annie, even though she's really N^3 (Zileas forgot Magic Penetration).

You are looking at sustained teamfight damage.

If you spend 1000 gold on AP, Annie gains N damage. If you spend 2000 gold on AP, Annie gains 2N damage. Etc.

If you spend 1000 gold on CDR, Annie does her damage more often, resulting in N more damage. If you spend 2000 gold on CDR, Annie reaches the cap, but is doing 2N more damage.

If you spend 1000 gold on AP AND 1000 gold on CDR, you get multiplicative results.

For increasing AP:

Total damage = Base damage * (1 + N). N is typically a fraction relative to base damage (and you could say that since Annie has a total AP ratio of 1.9 on offensive abilities, she requires 1035 damage, or 544 AP to double her damage. Therefore 1000 gold is ~50 AP, or an N value of .09)

For increasing CDR:

Total damage = Base damage * (1 / [1 - CDR]), with cdr expressed as a value from .00 to .40. At 40% CDR, you are doing 66% more damage. Since there are no flat CDR items, I'm going to use Kindlegem for math. For 375g upgrade cost you get a minor amount of "free health", and 10% CDR, so I'm valueing 10% CDR at 375g. 50% CDR would double your base damage, and 1000g is worth 26% CDR. So 1000g would produce an N value of about .52.

Therefore, if you spend 1000g on AP, your damage goes up 9%. Meanwhile, if you spend 1000g on CDR, your damage goes up by 52%. Now, these numbers aren't raw game value, as burst often is more important than sustained damage for a mage, especially given the high damage yield of Annie's AoE potential and long cooldown on Tibbers.

However, the point is that if you spend 1000g on AP *after* getting your CDR, you get more damage out of it than if you didn't have any CDR. This is because they are multiplicative.

1.52 * 1.09 > 1 * 1.09

Now, let's move on to carries:

Level 18 damage is about 100, with attack speed about 1.0. No base armor penetration, typically minimal/no critical chance.

1000 gold gives 25 damage, 40% attack speed (about .3), or 20% critical. Additionally, it scales off Armor Penetration and Critical Damage.

So...

25 damage is a 25% increase.

20% crit is a 20% increase (that caps at 100%)

40% aspd (.3 speed) is a 30% increase.

If you have 6000g to spend, you can get:

150 damage (250% yield)

120% crit (conceptually a 220% yield, but caps at 200%)

240% attack speed (280% yield)

OR you can spend it equally on all 3, and get

1 * 1.5 * 1.4 * 1.6 = 3.36. That's a 336% yield. 50% increase more than the best straight single stat, and 20% more damage than straight attack speed.

THAT is what N^Y is all about. The more stats that factor into your damage that are multiplicative with each other, the higher you can spike your damage.

Champions like Twitch effectively begin with one stat (attack speed) capped out, and thus benefit more from other stats, causing them to scale faster.