League of Legends Guide
Minion count information  W.I.P.
written by skarnicaTable of Contents
Max amount of farm you can achieve in 10 minutes  1 
Max amount of farm you can achieve in 20 minutes  2 
Max amount of farm you can achieve in more than 20 minutes  3 
Introduction to farming gold  4 

Max amount of farm you can achieve in 10 minutes
Minion waves spawn every 30 seconds (6 minions)
Minion waves with siege cannon minion spawn every 3 waves (7 minions)
Minion waves spawn AFTER 1 minute and 30 seconds into the game (1.5 minutes)
1.5 minutes = 3 waves, thus
10 minutes normally = 20 waves (3) = 17 total waves
Out of the 20 waves, there are 6 waves that contain 7 minions (waves 3,6,9,12,15,18)
However, wave 3 is gone because it doesn't exist (minions spawn at 1.5 minutes in) > Waves 1, 2, and 3 are gone
20 waves (total)  6 waves (7 minions) = 14 waves (6 minions)
14 waves (6 minions)  2 waves (first minute of game) = 12 waves with 6 minions
6 waves (7 minions)  1 wave (last half minute of 1.5 minutes of game) = 5 waves with 7 minions
Result  10 minutes = (12 waves x 6 minions)+(5 waves x 7 minions) = 72+35 = 107 farm (minions)
So in 10 minutes, the max amount of farm you can achieve is 107.

Max amount of farm you can achieve in 20 minutes
Easy to calculate using the previous note; felt like adding to make it more convenient.
For anyone who asks: No, you cannot just add another 10 minute wave calculation to the first note because there is a calculation error in between, you'll end up being off by 1 minion count.
20 minutes  40 waves
However, we know that 3 waves are gone from the 40 waves (2 of them being in the 6 minions wave, and 1 being in the 7 minions wave)
40 waves consist of 13 waves that contain 7 minions.
Thus, 40  13 = 27 waves that contain 6 minions
27 waves  2 "faulty" waves = 25 waves x 6 minions = 150 minions
13 waves  1 "faulty" wave = 12 waves x 7 minions = 84 minions
Therefore, in 20 minutes, the max amount of farm you can achieve is (150+84) = 234 farm.

Max amount of farm you can achieve in more than 20 minutes
This is where it becomes a little bit different.
When the timer ticks past 20 minutes, the siege cannon minion waves comes every OTHER wave (once every two waves)
Thus, it's more frequent than the first 20 minutes of the game.
Therefore, we must add on our first 20 minutes of the game max farm onto whatever is after.
Formula = 234 farm + (what we will calculate now)
Since there will be 7 minions every other wave now
Therefore, 10 minutes = 20 waves = 10 waves of each (6 and 7)
So 30 minutes into the game = (30  20) = 10 minutes AFTER the first 20 minutes
20 minutes + (10 minutes AFTER 20 minutes) = 30 minutes
234 farm + (10 minutes)
234 farm + [(10 waves x 6)+(10 waves x 7)] = 234 farm (first 20 minutes) + 130 farm (10 minutes after)
I have conjured up a formula to calculate the max amount of farm after this 20 minute mark which is...
234 + 13(x20)
Where 234 is the first 20 minute farm, the 13 is the fixed variable, the x is the total amount of time, and the 20 is the first 20 minutes.
Examples:
30 minutes  calculated up there
234 + 13(3020) = 234 + 13(10) = 234 + 130 = 364 farm
40 minutes
234 + 13(4020) = 234 + 13(20) = 234 + 260 = 494 farm
56 minutes (to prove it works with other integers as well)
234 + 13(5620) = 234 + 13(36) = 234 + 468 = 702 farm
To prove that it is 702 farm, let's calculate it stepbystep.
20 minutes  234 farm
36 minutes after  72 waves total  36 waves of each  (36 x 6) + (36 x 7) = 216 + 252 = 468
234 + 468 = 702 
Introduction to farming gold
Summoner's rift
Melee minion gives 22 gold on its first wave, 23 gold on its second wave, then it increases by 1 every 5 minutes (10 waves).
Caster minion gives 16 gold on its first wave, 17 gold on its second wave, then it increases by 1 every 5 minutes (10 waves).
Siege minion gives 27 gold on its first wave, and it increases by 1 every 23 minutes (46 waves)