[georgefox]

Releasing any data or statistic based on sensitive data--even once--bears a privacy risk. The primary purpose of differential privacy is to quantify that risk, both for a single release of data and over many releases of data.

As for the number of analyses you can run, that depends on what you mean. You're right that differential privacy won't allow you to set up a database of _confidential data_ that can be arbitrarily queried infinitely many times with any meaningful privacy guarantee, but this is in no way unique to differential privacy.

What you can do with differential privacy is release noisy statistics once and let researchers use those statistics for arbitrarily many analyses. This is what the 2020 US Census is doing, for example.