Rummy has players trying to turn hands of seven or ten cards into groups of three or four sets of card that are either three/four of a kind or running flushes.
If you have a valid group you may play to the center of the table. Once you have played to the center, you may also add cards from your hand to other groups on the table.
The first player to shed all their cards wins.
This game is well covered by the tile game Rummikub, where more on table play is encouraged, but the tiles equate perfectly to two packs of cards so no expenditure is necessary.
I love the mathematically-inclined fun of completely breaking up tale groupings and reforming (so long as you leave valid groups of at least three cards) to be able to add one pesky card from your hand.
Good for families.