Multiplayer FPS Suggestions:

Tribes 1
Tribes 2
Planetside 1 and 2
Natural Selection 2
Overwatch
Paladins
L4D2
Lead and Gold - Gangs of the West (pretty much LAN only these days)

Co-op Suggesitons:

Vermintide 1&2
Deep Rock Galatic
Sanctum 1&2 (same devs as Deep Rock)
Space Hulk Deathwing Enhanced Edition
Portal 1&2
Warhammer 40k

Single Play Suggestions:

Thief 1&2
Beyond Good and Evil (great story)
Pscyhonaughts (great story)
No One Lives Forever NOLF

Puzzle Game Suggestions:

Hook
Klocki
Limbo