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

1. Nothing is free

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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.

2. see item 1

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Perhaps the question should be reframed.

3. see item 2

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