In theory you could rely on others having the kit but that's a)rude and unprepared b) relying on them having decent kit that won't hurt you when it breaks (I've seen guys buying D-shackles in Bunnings for "recovery". Like Bird I won't lend stuff out, and if I am doing a recovery, it is under my rules, with my kit that I know the condition of.
What Bird said but my "absolute minimum" would be:
Tyre gauge
Shovel
(9 times out of 10 letting air out of your tires and a bit of digging and reverse gear will get you out unless you've dug yourself down to the axles)
(For that other 10% and if bogged on the beach on an incoming tide
)
Rated recovery points
Rated Bow shackles
Winch Extension strap over a Snatch Strap if you can't get both. (both if you can get them but can you can use the Winch strap as a tow rope if needed - don't tow with a Snatch Strap. IMO too many recoveries go straight to snatch straps when other safer methods should be tried eg more air out of the tires, forward-back-forward back, digging out with shovel, gentle tow with snatch strap. )
Damper for straps - (can use a blanket or similar if required)
If you get a snatch strap get the knowledge to use it safely
(Compressor?: Expensive you can wait to the mainland to re-inflate if you really need to, just drive slow)