Not including the very basic knots (reef knot, overhand knot, etc that nearly everyone knows. Otherwise overhand knot, reef knot, and half-hitching would dominate every list based on real world use.)
1: Alpine Butterfly. It's a loop. It can be easily made into a bend. It's strong. It's easy to tie. It can be tied in the bight if needed. It can be tied around an object for use as a hitch. Double and triple loop variants are easy too.
2: Double Dragon. Same reasons as the Alpine Butterfly really.
3: Adjustable grip hitch. Neither the Double Dragon nor the Alpine Butterfly provides a good slide-and-grip knot. This does, and is strong and easy to tie.
4: Constrictor knot. Easy to tie. Makes a good hitch, a good binding, can be used as a whipping.
5: Figure-8 knot. Bring the end back around and follow itself for a good loop. Join 2 for a bend. Use it as a stopper. VERY easy.
6: Highwayman's hitch. Simple quick release. Easy to remember, if not terribly reliable. None of the above are quite as good for quick release, even when tied with draw loops.
7: Icicle hitch. Grips nearly anything. It can also be used as a prusik knot.
8: Splicing. Basic splicing is quite useful, and very strong.
9: Plaiting. Twisting cord into rope is quite difficult with more than two strands. If you need a thicker rope a good braid can supply one.
10: Turk's Head knot. With knowledge of expansions and interweaves you can create most of the decorative knots you'll ever need.
Really, one could get away with the Butterfly family, icicle hitch, splicing, and plaiting. All of the above are strong (well, except the higwayman's hitch). All are easy to tie. All are part of a larger series of knots, and can be used for multiple purposes.