A long, unsorted list of good topics to learn:
(Rota's) umbral calculus
model theory, abstract model theory
binomial coefficient congruences
bezout theorem
resultants
grobner bases
sturm chains
real closed fields: Artin's theorem on sums of squares of rational functions, Tarski's decidability results, positivstellansatz
Majorization: Karamata, Fuchs, Schur convexity
*completeness theorem*
universal algebra
cross ratios
hyperbolic geometry
LLL
hashlife
sturmian words
expander/ramanujan graphs
modular forms, hecke operators
quaternion arithmetic, and more generally arithmetic in division algebras
Matiyasevich's theorem, Waring's problem, the S-unit equation
nonsingular cubic surfaces
Multiplicative functions: Dirichlet convolution, Mobius inversion, average size of a multiplicative function, analogy between L-functions and generating functions, relationship between 1/zeta and the mobius function, ...
p-adic analysis: Hensel's lemma, "power lifting", primitive roots, Chevalley-Warning theorem, Newton polygons, Weil bound for elliptic curves over finite fields, Hasse-Minkowski local global principle, p-adic power series (for example, the p-adic exponential and p-adic logarithm)...
Reciprocity laws: Quadratic reciprocity, Gauss sums and Jacobi sums, Cubic reciprocity, Quartic reciprocity, ...
Point-set topology: image of a compact set under a continuous map is compact, Heine-Borel, ...
Convex sets: Helly's theorem, Minkowski sums, Brunn-Minkowski, ...
differential galois theory
shannon switching game
o-minimality of the theory of reals plus exponential function and tarski's exponential function problem and schanuel's conjecture
method of interlacing polynomials
nonstandard analysis
uniformization of Riemann surfaces
tropical geometry
quantum computation
schoof's algorithm for square roots mod p
proof theory, resolution proof system, extended resolution, frege systems, natural deduction, applications to SAT-solvers (minisat, glucose, ...)
finite projective planes
multi-armed bandit problem, upper confidence trees, application to go playing programs
Nielsen reduction
stable polynomials, positive definite matrices, completely/absolutely monotone functions
matroid intersection theorem and its applications
linear programming, semidefinite/convex programming
newton polytopes and irreducibility
spectral theorem
forcing (logic)
finite topological spaces
"frozen boundary" in tiling problems
mating post-critically finite rational maps?
cayley-menger determinant
percolation
Glauber dynamics and the Gibbs measure
viscosity solutions to pdes
graver bases??
context-free grammars
markov's inequality, chebyshev's inequality, chernoff bounds
maximal inequalities
banach-tarski paradox, measure theory
nevanlinna theory
gauge integral
Yang-Baxter equation, quandles, quantum integrability
model categories, small object argument, simplicial sets, Dold-Kan correspondence
queueing theory