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@chrisburton seems to be a Type sensei…looking forward to an answer.
my guess would be as many as fit the design naturally, without using fonts just to use fonts. So if you could use 4 fonts, and have it still look natural, it is probably okay. I like to use two, having headings and other special fonts match the logo, and the rest be the main body text. this is just my guess, since I am far from an expert. expect some far better answers incoming :)
I would use whatever will make your design look good while not hindering performance. If you’re embedding fonts or pulling them from a resource such as google web fonts or typekit, you need to be careful not to overdo it as the more fonts you add the bigger the download size.
Also be aware of your audience and the platforms they using. If you need to still need to cater to IE7-8 be aware that some embedded fonts can have rendering problems/failures in those browsers. Make sure you test and workaround them, as workarounds do exist.
Finally, you didn’t distinguish between multiple font families or multiple font weights. You could pick one good font family and load different weights (Light, Book, Bold, Black, etc) instead of choosing several entirely different font families.
Try googling “font combinations” for some inspiration.