server-start with dynamic server-name

How to determine if serverN is already launched?

server-running-p is a compiled Lisp function in ‘server.el’.

(server-running-p &optional NAME)

Test whether server NAME is running.

Return values:
  nil              the server is definitely not running.
  t                the server seems to be running.
  something else   we cannot determine whether it’s running without using
                   commands which may have to wait for a long time.

You can replace the call (server-start) with following code in your init.el:

(require 'server)

(defun my--server-start ()
    (let ((server-num 0))
        (while (server-running-p (unless (eq server-num 0) (concat "server" (number-to-string server-num))))
            (setq server-num (+ server-num 1)))

        (unless (eq server-num 0)
            (setq server-name (concat "server" (number-to-string server-num))))
        (server-start)

        (setq frame-title-format server-name)))

(my--server-start)

Note: to connect to those servers, you have to use emacsclient like this:
emacsclient -s server<SERVERNUMBER>
Example: emacsclient -s server2