Scaling up can be achieved trough many techniques
- Having multiple instances: aka Elastic Load Balancers
- Sharding: server 1 handles requests for users A-M server 2 handles requests for users N-Z
- Add caching: Are you servicing the same request multiple times? Throw some memory at the problem at keep serving the same answer
- Simplify your workload!
The really important question you need to answer is what is limiting your ability to server N+1 clients. Are you running out of sockets, memory, cpu time, db transactions?
Like any profiling problem work out what your dominant problem is and solve it.