Following some basic assumptions that you will have few users (probably less than 100) and the data you’ll get from the analytics services will not require complicated, long-running data calculations, will you be needing a large server?
On the contrary, you can start off with $5/month digital ocean servers and grow (horizontally) as you go.
If you want to host a server on premise and use an actual physical server, the smallest possible one you’ll be able to find will most likely hold up just fine for a long time.
But I may be wrong due to the assumptions I make. After all, the amount of computing power you require depends on the type of your app and the kinds of workflows it will serve and the type and amount of load the users will impose.