If you want to study graph theory or combinatorics [1], then calculus will be pretty much useless to you, and you'll naturally go years without using it.

Calculus is also useless in some situations in abstract algebra (which are said to have combinatorial character). There are other parts of abstract algebra, e.g. Differential Galois Theory [2], in which calculus is pretty important.

Topology is similar. Elementary topology is part of the foundation supporting calculus, while algebraic topology is one of the tools that's useful when we try to do calculus (or solve differential equations) in non-Euclidean spaces.

Fields making heavy use of calculus include differential geometry, differential equations (ordinary or partial), dynamical systems or control theory. That subsumes most of physics. Fields underpinning (and largely inspired by) calculus include real and complex analysis, measure and integration theory (aka axiomatic probability theory). Also functional analysis, which is a generalization of linear algebra, which is the bookkeeping methodology of calculus in higher dimensions.

[1] The first sentence here says it all: http://en.wikipedia.org/wiki/Combinatorics

[2] http://en.wikipedia.org/wiki/Differential_Galois_theory

> Many pure math classes require no (or very little) calculus.

These are not the same thing (hence my confusion.) Your initial comment seemed to indicate that nobody does analysis anymore, which is just not true at all (look at the most recent fields medal.)