Recommendations for On-Hand 74xx (and others?) ICs
A lot depends on what you want to do.

I like to keep a fair amount of glue logic around. My most common use parts are '00 (NAND), '32 (OR), 08 (AND), '04 (NOT), '06 (open collector inverters) and '14 (inverters with schmitt trigger inputs) and I really like the SN74LVC1G99 configurable logic gate from which you can implement many boolean functions.

I do keep a few bus latches / drivers around for interfacing projects. ('244. '245, '373 and the 16 bit variants).

There are some items apart from that, but those are the parts I personally use the most.

There will always be a few parts I do not have to hand, but I can usually get them within a day anyway.