SSE lets you keep the connection as long as you want. I am not sure what you mean by client-side. If you mean Web Browser then I believe there is certain limit of parallel HTTP connections that you can have depending on the browser. http://stackoverflow.com/questions/985431/max-parallel-http-....
I'm more wondering about the number of open HTTP connections that a server can support. Googling around, I see mentions of 64,000 for vanilla servers, perhaps millions depending on the server (affected by # of processor...?)
So if your SSE connection is 10 minutes long and you might have more than 64,000 simultaneous connections to your vanilla box in those 10 minutes, you just need to start load-balancing I guess.
There's a common misconception about the number of TCP connections a server can handle being limited by the number of local ports (~64K). The real limit is in the number of LocalIP+LocalPort+RemoteIP+RemotePort pairs. If your traffic is coming from arbitrary Internet addresses, then you are not limited by ports.
Of course you'll be limited by other factors such as kernel fd limits (which are adjustable) and the CPU needed to monitor lots of sockets.
The issue with maximum open connections will go away with HTTP/2, and you can get around it today using SPDY since both multiplex connections to a domain.
So if your SSE connection is 10 minutes long and you might have more than 64,000 simultaneous connections to your vanilla box in those 10 minutes, you just need to start load-balancing I guess.