The intuitive thing is that extra capacitance, but capacitance is the most difficult load of all for stability.
I will try and keep it simple. Power supplies use feedback to regulate the output which means you have a high gain amplifier with negative feedback. The negative feedback will change the output voltage until the difference between it and the reference voltage is zero.
Now add frequency to the equation. Due to internal capacitance, amplifiers loose gain as the frequency increases. What goes along with that is phase shift. If at any frequency a control loop has a gain more then 1 and the total phase shifts reaches 180 degrees, the control loop will oscillate.
Opamp designers have known about this forever. What they do is to make the input and output stages of opamps really fast, and then slow the middle gain stage down with a single RC time constant. This has a phase shift of 90 degrees from about 10Hz up to 1MHz (or whatever the opamp goes to). With one single RC dominating, the phase shift through the opamp's whole frequency range is around 90 degrees so it is stable.
Just to make this clear, if you had a 1KHz signal going into a unity gain opamp, the output would look pretty much in phase with the input. But if you looked at the microvolt AC signal between the inverting and non-inverting inputs, that signal would be leading by 90 degrees. Simply to have a zero phase lag 1KHz coming from the output, the actual input signal between the input pins has to be leading by 90 degrees.
Now we come to power supplies. you still have the opamp with its 90 degrees phase shift, but you have extra stuff too - power transistors, driver transistors, protection circuitry, etc, and no matter how hard you try, these all add extra phase shift. Ok so say you keep all these other things to an extra 45 degrees - the supply is stable?
Unfortunately for the designer, stupid people actually want to put loads on these power supplies. What happens when you put a massive capacitor on the output of a supply? You have another 90 degrees phase shift and it oscillates. The bigger the capacitor, the more likely the supply will oscillate.
So the power supply designer and the IC designer has to do the impossible - if you disconnect any filter capacitor from the output, the control loop has to have less then 90 degrees phase shift all the way from Dc to the 100kHz+ bandwidth of the loop. You can follow the opamp trick of making everything blindingly fast except for one RC constant and then you do some tricks to slow the roll of to less then the 6dB per octave of the RC time constant, and the result is you get a little less then 90 degrees. This is the technique probably used in the LT3080.
For the discreet componect supply designer, you usually want to use an opamp that has the 90 degrees RC time constant built in, so you have to do tricks with external RC networks to cause a bit of phase lead to cancel out the excessive phase lag. It is not easy which is why Dave went for the LT3080 solution. If he did design a discrete supply, he would probably be on video blog number 23 by now - still fiddling with the compensation.
The conclusion is that for most power supplies, if it is stable with a 100,000uf capacitor under all output voltage and current conditions, it is probably going to be stable for most other loads. Lots of power supplies out there are probably not stable with a huge capacitor, as the voltage is hardly changing on the capacitor - it is the control circuit in the supply going nuts. Most people wouldn't notice. This may seem OK, but it is not, as it will affect regulation, the current limit control, and a few other things.
Richard.