Ten.

Here's the list: https://k1.spdns.de/Vintage/Sinclair/82/Peripherals/IF2%20Cartridge%20Interfaces/Sinclair%20Interface%202/

And another: https://spectrumcomputing.co.uk/entry/1000193/Hardware/ZX_Interface_2

Broken down by date: https://k1.spdns.de/Vintage/Sinclair/Software/ZX%20Spectrum%20Software/Interface%202%20Cartridges/

And another: http://rk.nvg.ntnu.no/sinclair/computers/peripherals/interface2.htm

Here's some info about 3rd party diagnostic ones and so on: http://www.fruitcake.plus.com/Sinclair/Interface2/Cartridges/Interface2_RC_Cartridges.htm

Ten:

• Backgammon (Psion)
• Chess (Psion)
• Cookie (Ultimate)
• Horace and the Spiders (Melbourne House)
• Hungry Horace (Melbourne House)
• Jetpac (Ultimate)
• Planetoids (Psion)
• Pssst (Ultimate)
• Space Raiders (Psion)
• Tranz Am (Ultimate)

You’ll find more information on the Sinclair Interface 2, Interface 2 cartridges (by date), Spectrum Computing’s page on the ZX Interface 2, and Planet Sinclair’s.

Various third-party diagnostic cartridges etc. were available too.