Exponent[expr, form] gives the maximum power with which form appears in the expanded form of expr.
Exponent[expr, form, h] applies h to the set of exponents with which form appears in expr.
Example: Exponent[x^2 + a x^3, x] .
The default taken for h is Max.
Example: Exponent[x^2 + a x^3, x, List] .
form can be a product of terms.
Exponent works whether or not expr is explicitly given in expanded form.
See The Mathematica Book: Section 1.4.7 and Section 3.3.2.
See also: Coefficient, Cases, IntegerExponent.