There is no definitive count, but estimates range from hundreds of thousands to over a million. Here's why a precise number is elusive:
If "board game" means commercially published tabletop games with a board, the figure is likely 150,000–250,000. Including all tabletop analogs (cards, dice, tiles, RPGs) pushes it toward 500,000+. The number grows daily.
For researchers: BGG's API and the The Game Crafter catalog offer the most searchable datasets.