What ways can you imagine? Within a few minutes I came up with:
1) Someone else paying for you seeing ads.
2) You paying for using the site or an associated service.
3) Someone else paying for influencing what the site does for you.
4) Someone else paying for your data on that site.
And, the one that people rely on too much:
5) Someone else paying for nothing, hoping the site can be sold later at a profit.
One that can realistically cover only a small portion of sites:
6) The site being run as part of a charity.
Do you find one of those generally more preferable than 1), or can see a point I'm missing?