My favorite is the Koike-Norton-Zagier product identity for the j-function (which classifies complex elliptic curves):
j(p) - j(q) = p-1 \prodm>0,n>-1 (1-pmqn)c(mn),
where j(q)-744 = \sumn >-2 c(n) qn = q-1 + 196884q + 21493760q2 + ... The left side is a difference of power series pure in p and q, so all of the mixed terms on the right cancel out. This yields infinitely many identities relating the coefficients of j.
It is also the Weyl denominator formula for the monster Lie algebra.