With CST-100 looking likely to start flying (eventually) and Crew Dragon prepping for a human spaceflight, plus the existing Soyuz program I got to wondering how many craft could be docked at the ISS at the same time.
Short answer: 8
For great info on available ports, check out this question on ISS Docking ports.
I will link in the awesome picture from that question here.
The Russian segment has 4 docking ports. The ATV is out of service, but that port is used by Soyuz and Progress, plus the other three ports.
The US segment has 4 ports. Where the HTV is berthed and the the Space Shuttle are docked have PMAs with IDAs added so that commercial crew can now dock there. (Starliner or Dragon).
Where the MPLM module is pointing down, is now a berthing port for Cargo Dragon/HTV/Cygnus vehicles. And below the HTV is the other berthing port for those vehicles.
Will we ever see more than 6 vehicles connected? The STS-133 mission shows how it looks with 6.
There was a mission earlier in 2019 where it was possible to have Cygnus and Dragon Cargo berthed at the same time, which would have gotten us back to 6.