When I was a teenager, my uncle gave me a calculus book and told me that mastering calculus was the most important thing I could do for starting out in math. So I learned the basics of calculus from that book. Later I read Michael Spivak’s two calculus books. I took courses that built on calculus, and studied generalizations of calculus such as calculus with complex variables, calculus in Banach spaces, etc. I taught calculus. After a while, I started to feel maybe I’d mastered calculus.
Last year I started digging into automatic differentiation and questioned whether I really had mastered calculus. At a high level, automatic differentiation is “just” an application of the chain rule in many variables. But you can make career out of exploring automatic differentiation (AD), and many people do. The basics of AD are not that complicated, but you can go down a deep rabbit hole exploring optimal ways to implement AD in various contexts.
You can make a career out of things that seem even simpler than AD. Thousands of people have made a career out of solving the equation Ax = b where A is a large matrix and the vector b is given. In high school you solve two equations in two unknowns, then three equations in three unknowns, and in principle you could keep going. But you don’t solve a sparse system of millions of equations the same way. When you consider efficiency, accuracy, limitations of computer arithmetic, parallel computing, etc. the humble equation Ax = b is not so simple to solve.
As Richard Feynman said, nearly everything is really interesting if you go into it deeply enough.