Not sure if these are really strategic but they do require brains.
Skip-Bo – but I like Phase 10 better.
Mahjong – it’s really expensive (used is around $100) but a lot of fun.
Rummikub
Malarky – is a great one for laughs. Not really Strategic and doesn’t have a board but you do need your brain to make up lies. (Which isn’t assur since everyone will know your lying right away. Which I learned is fine.)