Is this even realistic at Harvey Mudd College?
I think a few students double major in general, but the number is small. Either major is already a ton of work. There would be nothing stopping you from taking a few additional math classes beyond what is required, too, even if you don’t double major.
I think double majoring in math/physics or math/CS is doable. Double majoring math/engineering is very difficult just because the course load will kill you. Just majoring in engineering alone is difficult I think. I know someone who did double major math/engineering at HMC but he came into HMC with about 10+ transferred math courses and started his freshman year with Math 121 (Fund Math is what Mudd students call it - no pun intended) and only needed about 5 more math classes to satisfy the math major requirement. But nothing is impossible 
I don’t think Mudd probably actually took the math credits, but they may have allowed him to skip some of the core math sequence. But you have to test to do that at Mudd. It would be highly unusual for someone to test out of all of the 9 credits of core math, though – no one should expect this. And currently there is no Math 121 at Mudd, so not sure what this reference is.
Regarding some of the “double major” combos mentioned above, Physics offered some sort of ‘combined’ majors that are like Physics with a CS emphasis, for example. They allow substitution of some top level Physics courses with higher level CS courses. It isn’t exactly a double major, though. I don’t know if other majors have this kind of combination or not, but here is how it works for Physics (click on Options near the bottom of the page):
Thank you for all of this information @intparent and @tigerbookmark