If I'm not mistaken, emails offering free stuff are more likely to be considered spam by spam removal software. Is there a similar problem with offering free stuff on a website? Of course, if someone will be searching for "free cake", he'll be shown those results. But what if he's searching for "cake"? Will a site offering "free cake" rank lower because it will be considered spam-like?
If your site is useful, relevant, and does not participate in spammy practices, there should be no adverse effects for using the word "free". I have worked on legitimate erectile dysfunction (big pharma) sites, which is normally a spam-heavy subject, and they rank just fine. Now, if you are buying links, link to "bad neighborhoods", and cloak keywords - yes, you'll be penalized. Stay legit and you will be fine.
But what if he's searching for "cake"? Will a site offering "free cake" rank lower because it will be considered spam-like?
Google's objective is to serve the most relevant content as fast as possible to user queries.
So if user are querying for 'free cake', Google will serve 'free cake' content of high quality and skip 'free cake' content of low quality. In other words, the word 'free' has no positive of negative impact on rankings by default. Only the content determines whether stuff is spammy or not.
If users like your 'free cake' content and Google sees it, it may decide to show it in 'cake' search results too. And if 'cake' users like it too, it will rank there too on the long run.