For me, secondly it's all about the expanded options for both players and DMs. FIRST, that would be fixing anything I consider unbalanced/broken in the core.
1e: UA - while not exactly a balanced book and thus not fixing anything in core, no other book in 1e did fix the core and at least this expanded options for everyone (fills my second condition)
2e: Tome of Magic - Just edging out the PO:Sp&M since even though this didn't have the reorganized spheres, it does have the spells for the spheres and I'm going to cheat and use the DRAGON issue that was the inspiration for the reorganized spheres.
3e: I'm tempted to say PHB2, but I'm going to have to go with Tome of Battle: Book of Nine Swords as this one book fixed everything I thought was wrong with melee combat.
4e: No question about it, PHB2 is as core as you can get without being "core".
NOTE: If computer products were available, for 1e/2e, I'd change my vote to my DRAGON cd archive compilation and for 4e, my DDI subscription.