Thus far, peak at about 3000 REST requests per minute and 400 simultaneous ActionCable (websocket) connections. We're still scaling up the app, as we just launched it about a week ago, but we haven't yet hit the limits of the platform.
Actually just 1 Performance-M dyno. With Sidekiq for external/resource-intensive requests, and a pretty large pool configured in Passenger.
ETA: It doesn't get as much press, but ActiveJob is one of my favorite parts of Rails 5. It's an agnostic implementation of background jobs -- ActiveRecord : (Postgres, Mysql, etc) :: ActiveJob : (Sidekiq, Resque, etc). Very nice, flexible syntax and it helps make your application platform-agnostic.
Second ETA: As rapind pointed out, ActiveJob has actually been available since Rails 4.2.