I used to go to a bar in Santa Monica (Warszawa) where the Google folks go every Friday for happy hour. The recruiters kept trying to get me to apply, but I never did. I get the impression that they want people with Engineering/CS backgrounds and here I am a programmer with an English degree. I bet I'd have a better chance of building a startup and selling it to Google so that I could work there than going through the application and interview process. I could be wrong though...
I work at Google and I am an engineering-degree dropout who was referred by a music-degree dropout; we both now report to a VP who was a mathematics-degree dropout.
That's not to say it won't be harder to get past resume screens and such, and a referral does help in these cases. Thetrumanshow is right that Google suffers when people like you don't apply.
Strangely enough, being a dropout sounds sexier than having a BA in English! This all happened waaaay back in 2007 when my resume wasn't as impressive. Since then I've really fleshed it out, and I can get hired places on the strength of my accomplishments regardless of what I studied a decade ago. It's too bad working for Google doesn't sound as appealing now as it did in 2007, or I might just try applying.
For the record, I now work for ClearChannel (a pretty big company), and it is by far the best place I've ever worked. You do feel the size of the company when it takes 5 separate IT guys to set up your workstation (hardware, software, email, phone, VPN), but as far as the actual day to day operations, I couldn't ask for a better environment. I am the only non-CS dev on the team, but I can hold my own.
Sure, all generalization break down upon examination, but I think any company that uses five people to set up a basic cubicle is pretty much a direct opposite of Google in corporate structure and culture. There are indeed companies, and large ones, where the autodidact can find a home, but it requires the kind of organizational looseness that an overwrought bureaucracy brings. In Google terms, this would be a lowering of the bar.
FWIW, I'm mostly autodidactic (have a CS degree but got it mostly in my last semester, after learning all the CS stuff while blowing off my physics major) and don't consider myself particularly social, and I've found Google to be a pretty good fit so far.
There are a lot of dominant social and professional conventions that derive from university experience, and I was using autodidact with the intent of excluding college graduates (or at least technical ones) in favor of the more archetypal bedroom geek.
It very much depends on how social and how much of a self-promoter you are. If you are willing to fight for the latitude to follow your own direction you will do really well. But it is important to be social as well, as you'll encounter more opportunities as well as your own sharp edges quicker.
I had recruiters calling me for 9 months; including somehow trying to reach me through calling my parents house (i don't know where they got that info, but I guess it's on google... either that or a really really old resume). I started the interview process while working on a startup, but then cut it off in the middle and decided google would always be there but my 20's wouldn't.
