It depends enormously on the guarantees you want to provide to your customers.
If you don't advertise anything, then it should be sufficient to test only on a single platform.
If you state that it works on platforms X, Y and Z, then you better have the test reports to prove it.
And this is where it gets interesting. With PC applications, people tend to be far more accepting of bugs or incompatibilities than with other electronic devices, such as mobile phones. This is reflected in the testing effort and for that reason, major app developers test their app with such a large number of devices, just so they can account for any possible incompatibility.
As an anecdote, I have worked for a company making car-kits (so you can use your mobile phone in a car). Those car-kits get tested against just about every mobile phone that is on the market, because people just don't accept it if their phone does not interoperate flawlessly with their car-kit.