The ones I first worked out on my own were that all whole numbers were the result of the sum of a single set of only primes raised to powers (2^3 + 5 + 11^2) or the *product* of a single set of primes raised to powers (2 * 3^2 * 7). Then some teacher told me there was no chain rule for integration, and I found the chain rule essentially a mental mathematics strategy for solving complex derivatives, so I took the next 40 minutes to analyze several integration exercises and produced integration by parts (which we learned about a week later--what a waste of time). Simple stuff.
My favorite one was physics tensile problems. I *hated* tensile problems. To solve a tensile problem, we had to carry out a seven-step algorithm in which we'd break down each angle into its horizontal and vertical component vectors, then solve the right triangle for each, and combine the solution's horizontal and vertical vectors, solving for the hypotenuse.
In that picture, consider T1 and T2 as the length of those sides (they're the tension on each rope or whatnot they represent). M is the hanging mass. As it turns out, you can get a triangle by placing a line of length M between the top left point (where angle Theta is) and the bottom right vertex (where T2 meets the vertical wall); or by moving T2 *without rotating it* such that any of its vertexes connects to any of T1's vertexes, and then connecting the remaining two with a line of length M. I recognized this largely by mathematical result.
Pick a set. You'll either end up with two sides and an angle or two angles and a side. You can now glance at this diagram, apply the Law of Cosines, and solve it in one step. When I showed my physics teacher, he said he didn't see any mathematical reason that would work, although it *did* work on every problem we tried. Should have asked the Asian chick who took every form of math there was when she went to college; my teacher was largely a materials science type of guy.
Obviously, this one's my favorite because it's a *much* simpler way to tackle an irritatingly tedious problem *and* my academic superiors could never understand why it worked. That means I didn't waste my time figuring out some mathematical trick I could have found by flipping a dozen pages ahead in the book. As far as I know, this is a known technique, but *very* few sources mention using either the law of sines or the law of cosines to solve tension triangles.
This is why math was always fun for me. I reflected a lot on how it all fit together.