At the end of the two-weapon fighting chain, you would get three extra attacks. One from each feat in the chain.
He basically got them early. And they have a higher base attack bonus than a two-weapon fighter's offhand bonuses, as they are extra attacks based off of his first one, while two-weapon fighting attacks are based off of iterative attacks.
Iterative attacks occur naturally once you reach a certain bonus. +5 turns into +6/+1, with you being able to make an extra attack using the lower bonus. Every five bonuses thereafter adds another attack. depending on class, that could be a maximum of 20/15/10/5 without any two-weapon fighting feats.
Two weapon fighting imposes penalties dependent on weapon type. Light-classed weapons impose a lesser penalty for example. So at level one, with 1 base attack, a two weapon fighter would attack twice in a round at -1/-1, as there would be a -2 penalty.
The problem player has been allowed to get improved and greater multi-weapon fighting at the points where he would normally be able to get the two-weapon fighting versions. So a level 20 two-weapon fighter would be able to attack seven times in a round, with bonuses of +18/+18/+13/+13/+8/+8/+3. Problem player at level 20 will attack like this: +18/+18/+18/+18/+13/+13/+13/+13/+8/+8/+8/+8/+3
A total of 13 times. Per round.
Edit: Also,
here's what swarms can do, if anyone is interested in why a PC with swarm abilities is outright terrifying.