There are a number of ways. Lets say it is a 50% duty cylcle square wave at a frequency of 10 MHz. A square wave has odd harmonics.
1) it is easy to produce 30 or 50 MHz from the squarewave using only analog means. You would simply make either a 30 or 50 mhz bandpass filter, it would pass only the energy you desire, then you could use a transistor amplifier to reamplify the signal to make it useable. But since only odd harmonics are present, you are not going to get much 20 MHz.
2) You could lowpass filter the 10 MHz squarewave to turn it into a sine wave. You then would amplify it, input it into a transformer primary. At the secondary, you would ground the center tap, and use a series shottky diode on the + lead, and a reverse series diode on the -lead. You combine the two diodes together and you would get a fullwave rectifier, or "push-push" type of output that would have strong 20 MHz, and have the 10 Mhz somewhat suppressed.
3) using digital means, you can do some things with XOR gates and short lengths of coaxial delay line to take a single pulse and turn it into 2 pulses--which would look like twice the frequency. But that would be cheating because it uses digital circuits.
4) you could have an analog 20 MHz voltage controlled oscillator. You would sample its output with a diode sampler circuit at a 10 MHz rate. You would take that sampled output, filter it with some R-C components, and drive the tuning port on the oscillator. A sampler made from diodes is considered an "analog" circuit, so you would be legal.
5) you could drive the square wave into some highly non-linear device, like a saturated ferrite or a voltage controlled capacitor (varctor diode). As it charged the reactive device, the non-linearity would cause currents to flow at 10, 20, 30....MHz. You would filter off the 20 Mhz, and reamplifiy it.
6) you could drive the 10 MHz square wave into a length of transmission line. Along the length of the transmission line you could place some high impedance voltage sampling taps. The first tap goes to a + on a summing junciton op amp, 2nd tap goes to - on summing junciton, 3rd tap goes to + on summing junciton, 4th goes on - summing junction. End of transmission line is terminated in the line's characteristic impedance for no reflections. As the pulse travels by first tap op amp goes high, after pulse travels by 2nd tap op amp goes to zero, as pulse travels by 3rd tap op amp output goes high, etc. Need to space taps by the right time delay.
7) you could have some electromechanical mechanism that was tuned to 20 Mhz (like a crystal resonator, or a MEMS cantilevered arm, etc) and you could capacitively coulple the 10 MHz square wave into it. The spike from the leading edge of the 10 MHz square wave would inject energy into the resonant circuit, where it would "ring" at a 20 mhz rate. For that matter, you could probably just have an L-C resonator that you capacitively feed with spikes from the square wave input, and it would ring out at the 20 MHz rate.
Probably another 20 more ways if you think hard enough.