I altered the river action in this hand for the sake of providing an example.

But for the sake of argument, in hands such as this where we are almost certainly beat, and are facing a river bet for X amount of chips -- what is the maximum amount of chips we can call in order to gain information on the player?

EDIT
criticism on play up to turn is also welcome, but only after you answer my original question

Full Tilt Poker $30 + $1.50 Heads Up No Limit Hold'em Tournament - t10/t20 Blinds - 2 players
The Official 2+2 Hand Converter By DeucesCracked Poker Videos

BTN/SB: t2900 145 BBs
Hero (BB): t3100 155 BBs

Pre Flop: (t30) Hero is BB with 9 J
BTN/SB calls t10, Hero raises to t60, BTN/SB calls t40

Flop: (t120) 2 A 3 (2 players)
Hero bets t80, BTN/SB calls t80

Turn: (t280) 4 (2 players)
Hero checks, BTN/SB checks

River: (t280) 2 (2 players)
Hero checks, BTN/SB bets t205, Hero calls t205

???

calling off chips for such info isnt that great at any price. it doesnt rly matter if your J high loses to A, 55 or Q8s here. you could always try out some bluff raises instead of calling. if he call, u will see anyway, but they will fold quite a bit too.

there isnt a certain number of chips to call.. its about pot odds or if you know youre beat what your investment will be (if its worth it to you) to ONLY call to get information.

usually when i run across this example its in the first few stages of a husng and its a small river bet that is either a very small bet (ex: min bet for 30), or the pot is ridiculously big (ex: 600, and i have to call like 80 with 3rd or 4th pair or something)

as for ur play it looks pretty normal/standard before the river.