My basic essentials would be:

Round Turn and Two Half Hitches
Figure Eight
Hunters Bend or Zeppelin Bend (closely related)
Bowline
Constrictor Knot
Tautline Hitch
Square Knot

Other very handy knots:

Ashley's Stopper Knot
Double Fisherman's Bend
Angler's Loop
Alpine Butterfly
Ground Line Hitch
Simple Simon (under)

If you end up stuck with that ultra cheap super-hard, tough-to-bend three-strand-twisted slippery nylon junk rope, in order to get a knot to hold you need something more obscure like:

Vice Versa
Surgeon's Knot