Limit on number of users in postgresql?

I want to make a client application that logs onto a postgresql server
with an individual postgresql user role for each user.  This is as
opposed to having one user role for the entire app, storing that
password on the local client, and using the client app to do
authentication against a a table in the connected database.

I'm wondering if there's a manageable limit on the number of users in
postgresql.  I could have conceivably thousands of different users for
any particular app and multiple apps in the same server.  I would
distinguish between different users for different apps by something like
a user encoded name myapp_myname where to the individual app that would
be user myname.

This seems like a persistent problem for a lot of database apps.  Is
there any simple way of dealing with this problem?  A lot of apps store
root passwords to the database locally, which is very insecure,
especially on desktop applications.  It seems like a big headache to
declare users globally to the server instead of at the database level.
Am I doing more work than I need to do?

Thanks.