What are some things that the government could offer for free?

Everything that "government" does is paid for by taxpayers (and debt / inflation, a different form of tax).

So to answer the question: Nothing. Nothing the govt could offer can ever be free.

Perhaps the question should be reframed.

I would suggest re-framing as follows: "what are some things that the govt coerces taxpayers to pay for, that it really shouldn't?"