Hiya Muphex.
Its good that cost is no problem, because Amiga is an expensive hobby.
You have two choices if your goal is just games.
Amiga500 for strictly the old stuff. Dirt cheap mostly, but with this, I'd try getting an A500 that has some memory addition and the HD that fits in the side slot. Use the WHDload program as a lot of the old games are on floppy disks that are slowly degrading. If you can get an A2000 with a HD card at minimum, that can work too.
Amiga 1200, relatively cheap, and needed for playing games that require AGA, normally comes with a laptop size HD. Backward compatable with older games by pressing the two mouse buttons before booting up, and choosing OCS and nocache, although its recomended to get WHDload, mostly because it eliminates copy protection. All you need for a decent performance is one of the 68030 or 40 accelerators and a SIMM memmory stick.