Suppose one is asked to construct a regular $n$-gon, but with one extra operation allowed in addition to the standard compass-and-straightedge ones: trisecting any angle.
Are all $n$-gons constructible now?
If not, which ones aren't?
What other operations do we need to add to our "toolbox" to make it possible to construct $n$-gons for all $n$? Of course, I suppose allowing $n$-section of angles for all integers $n$ would make it easy, but can it be done with less?
What you can do if you have an angle trisector:
Construct a regular $n$-gon when n has no prime factors greater than 3.
Construct a regular $n+1$-gon for any of the above values of $n$ if $n+1$ is prime. Such primes are called Pierpont primes.
Construct a regular $m$-gon when $m$ is a product of distinct Pierpont primes, a power of 2 and a power of 3. The powers of 2 and 3 could be $a^0 = 1$.
For example $13 =12+1$ is a Pierpont prime, so that's good, but $11$ does not fit any of the possibilities given above. (The regular $11$-gon can be constructed with the more powerful tool of neusis; currently this is the only known prime covered by neusis but not by Eucliean tools plus angle trisector alone.)