Theoretically, there are infinitely many map projections. They are usually defined by precise mathematical formulae, although so-called comprise maps (most notably the Robinson projection) also exist, which uses pre-defined values at specific latitudes and interpolation to calculate the exact projection.
I am more interested in the former - projections defined by mathematical formulae. For example, the famous Mercator projection is defined by the following formulae: $$x=R(\lambda-\lambda_0)$$ $$y=R\ln[\tan(\pi/4+\psi/2)]$$ where $\lambda$ is the longitude and $\psi$ is the latitude.
Now, there are many other "funky" map projections like the Euler spiral map projection and waterman butterfly projection below.
How can I go about creating my own mathematically defined map projection?
This does not have to be practical or anything, but I would just like to be able to create an original map projection since there are infinitely many.