The number of channels needed depends on what you're doing.
To take the example of the ATX power supply which Phaedrus brought up. For some purposes, it might be possible to monitor different rails separately. If you're just wanting to measure the ripple on each rail independently, you might even be able to get away with a single channel. But for example, what if you were interested in finding out how the other rails behave as you suddenly put a transient load on the 5V rail? If you had enough channels, the fastest way to do the experiment would be to probe all the parameters simultaneously as you added and removed the load on the 5V rail. If you only had two channels, you might be able to probe the 5V rail's load and then probe each of the other rails separately, one at a time, moving the probe around and repeating the application of a load on the 5V rail. If your time is worth much and you do that job often, you'd probably buy a four channel scope, but for an occasional hobbyist, you might eventually be able to get the information you needed with two channels.
Multiple channels are very useful when several things are varying at once, and interacting with one another, and you want to see the phase or timing relationships between them.
Fully characterizing an audio amplifier might require four channels. You might be interested in the input voltage and current, and the output voltage and current. That could be four channels right there. Or maybe you'd be interested in the voltages at a few intermediate stages along the amplifier, in addition to the input and output voltages. Or if you have a stereo system with some potential coupling or interaction between channels, you might be interested in left and right inputs and outputs. There are plenty of ways to use up four or more channels.
Suppose a microcontroller is sending and receiving serial data from a computer, and also producing analog PWM output, and maybe receiving analog input, and something's not working quite right, and you need to debug the microcontroller's firmware. It might be helpful to see all the relevant inputs and outputs on the same stream of data to see the sequences involved in the malfunction. When some signals are digital, maybe you don't need the full resolution of an analog channel. This starts getting into the area where a MSO or logic analyzer becomes useful. In digital systems, you may want many more than four channels. For example, it's sometimes handy to watch all the lines on a parallel data bus or address bus.
It's hard to say exactly how much of a premium four channels is worth over two channels. More channels are always better, but exactly how much better depends on the circumstances. I'll just point out that I have never heard anyone say, "My four channel scope is OK, but I just wish it didn't have so many channels."