I have read about people buying bitcoins by sending cash through the mail. That is perhaps the most anonymous one can get.

The seller has your mailing address. Mining is probably the most anonymous way to do it, and by far the least cost-effective. There are, however, ways to launder your bitcoins if you really want them to be anonymous, regardless of how you acquired them.

You can mine another cryptocurrency with your GPU and use that to convert to BTC. That's actually mildly profitable if you choose the right currency and your electricity costs are not too high.

If you are buying bitcoin, all you need is the seller's (seller of bitcoin) address to send the cash to.