Some (and I mean a very few) Amiga games/programs can be forced to run on a Graphics Card and thus bypass AGA.
You could try getting hold of ModePro from the Aminet. I've used ModePro to force certain programs to retarget to my Voodoo card, and thus having to avoid running two monitors at the same time.
I've had very mixed results so far though. Colonization runs fine, Civilization AGA can be persuaded to run (use the "Force Planar" checkbox in Modepro) but the sprites aren't visible and I'm relibaly informed that Sim City 2000 will run with "Force Planar" enabled. Non-game programs like the Magic64 emulator, Visage etc can also be used with ModePro. BTW, AFAIK a promotable patch for Alien Breed 3D II exists too.
As for anything that looks like it requires the Amiga's custom hardware (99% of games) then there's little you can do other than either re-routing via a TV card in the Mediator (I've not tried this so can't comment on success), keeping your old Monitor on standby or buying a scandoubler.
