List of Math competitions I encourage students to try:

Elementary School:

MOEMS (Elementary School)

Middle School:

Mathcounts
AMC 8
Purple Comet MS
Andover Math Open (Hard) (sometimes local)
Lexington Math Contest (Middle School) (sometimes local)

High School:
AMC 10/12, AIME,
ARML (need to find local coach for tryouts)
Purple Comet HS
USAMTS (proof based)
College contests if you are able: CMIMC, HMMT, Duke, PuMaC, etc.
Math Prize for Girls

All grades:
I have also heard good things about:
Mathleague.org
Math Kangaroo

Online:
Some AoPS Mock contests are good and very high quality. I like the CMC ones.
OMO
NICE
Tree Contest

Competitions that no longer run, but have good problems:
Mandelbrot
iTest

It is also great to try to make your own local math team or math club or math program. I highly encourage those trying to build a local math community to continue to try hard! Community is what keeps these things going, and having no local community in (insert endeavor here) makes it harder to succeed in said area.

Good practice:

Alcumus:
Mathcounts Trainer
FTW (for fun)
euclidea.xyz
AMC Trivial

Olympiads:
BMO/JBMO
USAJMO/USAMO
IMO ShortList
ELMO
CGMO
EGMO
Tournament of Towns
IMC (college)
Putnam (college, not olympiad)

The above all have good practice problems for olympiads. There are also geometry-only olympiads on AoPS forums.

Other countries:
Russian local olymipads have some good problems, same for some Eastern European Local olympiads. Other computational contests may exist in the UK, Canada, Singapore, India, etc, that may be at a good level for those preparing for AMC’s too.

Summer programs that I think are good

MathPath (middle school)
AweseomMath (middle and high school)
Canada/USA Mathcamp
SPARC
ROSS (NT Focus)
PROMYS (NT Focus)
Mathily
HCSSIM
RSI (11th grade)

Most of the above are for high schoolers

For latecomers to the math competition scene (i.e. starting in mid high school or as adults):
I encourage you to try some of the middle school math competitions, like AMC 8 or Mathcounts. Some of them are pretty tricky! Beast Academy also have some neat puzzles.

Other (non-Math)
USACO
F=MA, USAPHO
Advent of Code