I'm using ssh to connect to a server (really a cluster, but that's not important) and to transfer files to it. There seems to be a hard limit (9 in this case) of channels that can be used in one ssh connection.
Does the ssh protocol have a mechanism for querying the max number of channels available per ssh connection?
No, there is no way to query the maximum amount of channels available. The protocol is described in PROTOCOL.mux.
The server can limit the amount of multiplexed sessions (MaxSessions in sshd_config), but not the amount of the channels.