Max closed loop gain of an op-amp
Yes.  However, modern op-amps have VERY high open-loop gain, which makes the error (usually) acceptably small.  The caveat to this is that as your operating frequency goes up, the open-loop gain goes down.

A better way to express this is: an op-amp will behave ideally in closed-loop configuration so long as the open loop gain is >> than the closed-loop gain at the frequencies that you are operating at.