Is there a limit on the number of concurrent connections per server for Shiny Server?

No. Shiny Server is constrained only by its features. You can compare the features of Shiny Server Pro and Shiny Server here:  http://www.rstudio.com/products/shiny/shiny-server/

We price Shiny Server Pro by concurrent users per server to align what customers pay with the value they receive (apps that their customers really use), rather than other common measures such as the number of applications or “size” of server. See here for more on this: https://support.rstudio.com/hc/en-us/articles/220760228