Ach. There are so many.

Assuming you use the word "between" correctly, 1 and 10 are not themselves eligible.

1. 2 (the only even prime! the only solution to x+x=x*x=x^x, smallest sum of two positive squares and two positive cubes and...)
2. e (the amount of money I would have if I invested $1 for 1 year in an account that paid 100% annual interest, compounded continuously)
3. tau (Hope my old boss doesn't see this. He was an advocate for the legislation to see pi day recognized by the national government.)
4. phi (Just be careful of the pseudo-mathematics surrounding it, like seeing the ratio in finger lengths. It's an important number that shows up all over nature. But people tend to see it everywhere, whether it's there or not.)
5. 4 (first true composite number)
6. 6 (smallest number with two nontrivial distinct factors)
7. The square root of 2. (The length of the hypotenuse of the unit square; legend goes that people were once put to death for claiming its existence.)
8. 1.01001000100001000001000001... (a favorite example of an irrational number)
9. 9 (smallest sum of two distinct positive cubes)
10. 1 + epsilon (Just pick a hyperreal number that makes that precise. I mean any well-defined approximation to 1.0000...0001)

Hmmm.... my list if 40% rational numbers. Clearly shows some bias on my part, as the rational numbers make up precisely 0% of the numbers between 1 and 10. 

But then, they are everywhere. If you throw a dart at the number line, you'll never hit a rational number. But you'll always be really close.

No, even closer than you're thinking.