An oscilloscope is a crucial tool for serious circuit hacking. The minimum serious oscilloscope input bandwidth is 20MHz, which is to say, it attenuates 20MHz signal components by 3dB (and probably phase-shifts them), 40MHz signal components by 9dB (and phase-shifts them more), 80MHz signal components by 15dB, and so on; so on a "20MHz" oscilloscope you should still be able to see if there's an 100MHz signal, even if you can't measure it very well or see what its shape is.
It would be nice if it were possible to improvise a 20MHz oscilloscope out of inexpensive electronics.
Suppose, though, you're trying to work with an AVR ATMega328. This chip can only run at a max of 20MHz clock, and its ADC can only manage some 77000 conversions per second, and those at 7.5 bits of resolution. It would seem that you're pretty much fucked for seeing anything above 35kHz, which is some 500 times slower than you would like, and 500 times slower than the processor's own clock. That is, you're not going to diagnose much of anything related to an AVR with another AVR, right?
Well, consider this. It's common, although far from universal, for the 1MHz or 3MHz or 10MHz signal you're interested in to repeat many times. So if you can figure out its period, you can sample different points from many waves, then reassemble them into a picture of what one period of the wave looks like. Each sample-and-hold cycle should take place in well under a microsecond, and a faster external sample-and-hold circuit can be built, perhaps with an op-amp on its input to raise the input impedance.
(You probably can't show more than about 1000 samples horizontally on the screen at a time anyway, and you can't do that more than 60 times a second.)
Suppose, though, that it doesn't repeat many times? There's not a whole lot you can do about that if you don't have some way to store the analogue signal until you have a chance to digitize it. All the options I know of (Williams tubes, magnetostrictive delay lines, acoustic delay lines) seem improbable.
As for guessing the period, you can bandlimit your input signal with an analog input filter, sample the waveform at more or less random intervals, and then seek the period that provides the best continuity.
That won't give you a 20MHz oscilloscope but it should give you a 200kHz oscilloscope. One order of magnitude down, two more to go.