So I've only ever played through Kingdom Hearts 1 and 2. I just recently bought 2.5 and DDD, and I am either going to pick up 1.5 and the 2 ds games (358/2 and re:coded) and do a full series playthrough, or just play BBS and DDD and watch the cutscene movies of the other 3. Are 1,2,BBS and DDD enough, or is it worthwhile to play COM, 358/2 and Re:coded too?
From a story POV, I'd say the essentials are 1, Re:CoM, 2, BBS & DDD. Days has a few good moments but overall it's a whole lot of sea-salt ice cream eating. ReCoded is the most pointless story in an JRPG ever.
From a gameplay POV, ReCoM is very divisive. I personally like the card combat, others hate it or would at least prefer not to play it. Days is unremarkable from a gameplay POV, a kind of (even more) simplified Kingdom Hearts in many ways. It's not horrible by any means, but it's probably the weakest normal type KH, although its Sora development system is fun. I'd say 1, 2, BBS, DDD & ReCoded are all fun as far as combat goes, but they each have different shortcomings. 1 is a bit cumbersome and some of the level designs isn't all that good, 2 is a bit too straightforward though vastly improves the combat, BBS is (imo) the best game in the series from a gameplay POV but it did suffer from being on a handheld, DDD has the somewhat tiresome Pokemon stuff and the flowmotion stuff isn't super-well balanced and ReCoded is the millionth retread through a lot of the old worlds, even if its gameplay is probably the best that DS can handle.
I think playing through the whole saga in quick succession might be a bit much, so maybe play BBS, 1, 2 and DDD, watch cutscenes of ReCoM on Youtube, ignore the rest (unless you want easy trophies).