mongodb & max connections
You don't want to open a new database connection each time a new user connects. I don't know if you'll be able to scale to 20k+ concurrent users easily, since MongoDB uses a new thread for each new connection. You want your web app backend to have just one to a few database connections open and just use those in a pool, particularly since web usage is very asynchronous and event driven.
see: http://www.mongodb.org/display/DOCS/Connections
The server will use one thread per TCP connection, therefore it is highly recomended that your application use some sort of connection pooling. Luckily, most drivers handle this for you behind the scenes. One notable exception is setups where your app spawns a new process for each request, such as CGI and some configurations of PHP.
Whatever driver you're using, you'll have to find out how they handle connections and if they pool or not. For instance, Node's Mongoose is non-blocking and so you use one connection per app usually. This is the kind of thing you probably want.