Can you find examples of modular forms which take on values as e.g. algebraic numbers of degree n ? I'm interested in finding special classes of algebraic numbers particularly when n=3, they don't have to come from modular forms.
If you want special classes of algebraic numbers of degree 3, and they don't have to come from modular forms, I strongly recommend doing a search for "Shanks' simplest cubics".
Also, a Google search for $$\rm algebraic\ values\ of\ modular\ forms$$ turns up a few links that look like they might interest you.
EDIT: The $j$-invariant is a modular function with the property that if $\tau$ is a quadratic irrational with positive imaginary part then $j(\tau)$ is an algebraic integer. See the Wikipedia article on the $j$-invariant.
The cubics mentioned in the first paragraph above are of the form $x^3-ax^2-(a+3)x-1$ for natural numbers $a$. The fields they generate have many special properties, well-documented in the literature.