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So the total number of units that can fight in a battle for one side, including reinforcements, is higher than forty?
However, only 40 units per side can actually be on the field at any given moment (with "Control Large Armies" enabled), if you fight the battle in real-time.
This means that you will do deployment phase and battle with 40 units, and the remaining 40 units will only enter the battlefield as units rout, withdraw, or are destroyed. At no time in a real-time battle will there be more than 40 units per side on the field.
For auto-resolve, all 80 units per side take part in the calculation.
I'm going to want at least three stacks to attack Kyoto on Very Hard difficulty. I could put my mangonel and lots of missile troops in my first stack and then withdraw them when they've used up their ammunition, to allow melee reinforcements to arrive.
Also, consider having your cav loaded as reinforcements for field battles: cav can move much faster from the "red line" to the fight than infantry can.
Say, for example, I have four full stacks of 20 units: Stacks A, B, C and D. I lead with Stack A, and Stacks, B, C and D are reinforcing.
If I have large 40-unit-per-side battles enabled, I will start with Stack A, but what about the rest? Will I get a full Stack B from close to the start of the battle, and then units from Stack C when units retire? Or will I get reinforced equally from Stacks B, C and D, so near the start of the battle, I'd have a full Stack A, and a third each of Stacks B, C and D?
Back to your original question that doesn't look like it was ever answered, and months late:
The maximum number of reinforcing armies is 3, and the maximum number of units in a battlefield is 40 as you said. If you have 4 armies in position to reinforce, 3 of them will be selected (randomly it seems?) as reinforcing armies. I'm not sure that there is any logic behind which 3 armies end up reinforcing, but 3 is unfortunately the limit.
I learned this by attempting to train generals' agent exp on rebels. For those who don't know, you want generals to reinforce whenever possible because this is the only way to give agent experience to all participating generals at the same time. This limit means that 4 generals can gain agent experience from 1 battle if 3 are reinforcing separately. Additional generals can gain combat exp with kills if they're extra units in one of the 4 participating armies, but no agent exp.
As for the rest of the discussion:
When you enter the battlefield, if your 4 armies (attacker/defender and reinforcements) total unit count is less than 40, all units will enter the battlefield. However, if you have more than 40 total units across your 4 armies, then your entire main army will join, then units will be selected at random from each of the reinforcing armies to join the battle. Each reinforcing army will contribute as close to the same number of units as the other reinforcing armies as possible. You could think of it like the opposite of a dealer dealing cards where 3 players shuffle their hands and take turns passing cards to the dealer instead. It's like that. If at any point during the battle you lose a unit or it routes off the map, then the reinforcing armies will continue contributing units in the same manner. I'm not certain the units chosen to join the battle are actually selected randomly, but I haven't noticed any logic to it so far.
Hope that helps someone!