@Autumn I guess it depends on how you count them. I'm looking at our schedule right now, and some of the vaccines supposedly protect against multiple diseases. So they're kinda a multi-vaccine in 1. And they're also multi-dose vaccines, that need booster shots.
If I count the unique vaccines in there, it's like 7 unique vaccines.
If I count every time there's the word "vaccine" in the line, there's like 12 injections till the age of 14.
If I count every disease covered in those multi-vaccine injections as a separate vaccine, I'm getting around 41. Could be more if I miscounted or misunderstood something.

But that's just obligatory vaccinations. There's also a few optional vaccines (like for varicella), and that seems to be another 10-12 doses altogether.

Do note, I live in what would be considered a second world country. You might want to check with other European countries too, like France, UK, Germany, to see how a country more comparable to US does things.