How many circuits are recommended for this basement game room remodel?
I'm going to leave minimal for someone else and answer smart....

Put the lights on their own 15A breaker.

Put receptacles on a couple of 20 amp breakers.

If you MUST share (or think you must) put half the lights and some receptacles on one circuit (probably 15 amp unless you really want to run 12 gauge to lights) and half the lights and some other receptacles on another. Having lived with shared outlets and lights, I'm opposed to being left in the dark when a receptacle is overloaded. A lighting only circuit almost never trips, IME.

Being a basement I think you need everything to be GFCI protected at this point - either with GFCI breakers or with a GFCI as the "first thing in line" as it comes off the breaker.

Tester 101's answer to my question about lighting circuits did come up with a figure of 3 volt-amperes per square foot based on use as a dwelling for minimum power required. But minimums can be exceeded.