Soap - liquid (I prefer Dawn) and bar (I like to have Dove and Fels-Naptha)
Ammonia - Windex is fine if you don't want to play with diluting straight ammonia
Hydrogen Peroxide (like bleach but works on organic stains)
Bleach (I only buy Clorox because it is non-sudsing which means you can purify water with it in a pinch)
Borax based cleanser (although, I'm a fan of "Soft Scrub" too or old fashioned tooth paste in a pinch)
Steel wool and/or SOS pads (rarely used but handy to have)
Rubbing alcohol (not something I use often but handy to have)
I don't use vinegar mostly because weak acid doesn't clean very well and it is a poor disinfectant. Soap and water is fine.
Lots of micro-fiber cleaning cloths.
Magic eraser (optional - unless you have a toddler, in which case it is mandatory)
On top of this I have:
Swiffer wet wipes for mopping the floor - they are just easier
Pledge (I don't use this often on my furniture but it is great for cleaning plastics and patent leather)
Tung oil (this is the 'polish' we actually use on our wood furniture)
I also keep baby wipes around for quick clean-ups
I looked through my cleaning supply bin to see if there is anything I forgot. When we were in the apartment, I bought liquid toilet bowl cleaner because those toilets were disgusting. Now that we're on a septic system, I don't think I'll ever have a reason to use it again. I also have those disposable 'dry' dusters. I have no idea WHY I have them - maybe hubby bought them before I moved here. Again, not something I use.
Other than the usual - dishwasher detergent, clothes detergent, that's it.