[benrbray]

Can you give any examples of useful non-terminating programs?

It seems to me all useful programs are terminating (or, we truncate some non-terminating procedure by some convergence criteria). The question is only whether we have the logical tools to prove termination at compile time.

[ncmncm]

Programs running in microcontrollers routinely do not terminate; they run as long as there is power. Even many ordinary programs run until they are killed.

You seem to confuse algorithms, which by definition must terminate, with programs, which do whatever the hell you like (and sometimes things you don't).