Why does generating EUV frequency combs require femtosecond enhanced cavities? When generating extreme UV (EUV) frequency combs using high-order harmonic generation, why does the harmonic generation step need to be inside a femtosecond enhanced cavity for the comb to work?
More generally, I don't quite understand the concept of frequency combs, and what it takes for a light source to be a frequency comb. So: why must EUV combs be generated in such cavities? If the EUV is generated directly from a phase-locked laser (not in a cavity), would it still be an EUV comb?
 A: In response to the questions which you included in a comment above. There were lots of questions there, and it's usually best to try and formulate them into a single, more concise question, or to ask them separately. I would have included this as a comment rather than an answer, but it is longer than the character limit.
1) Whilst I don't usually like to reference wikipedia*, the article on frequency combs is a good introduction to what frequency combs are, so here it is (read the whole thing, there's an applications bit at the bottom).
"A frequency comb is a spectrum consisting of a series of discrete, equally spaced elements" according to the wiki page, and that's about as broad a definition as you can have. They arise whenever you have a regular, amplitude modulated signal in the time domain, e.g., from pulsed lasers (associated mainly with modelocking). If the laser pulses are emitted once every roundtrip of the laser cavity, and the roundtrip takes $12.5$ ns, then the frequency comb produced will have a frequency spacing of $1/(12.5\rm{ ns}) = 80$ MHz (in this case, the frequency comb spacing is equal to the repetition rate of the laser, and is formed of the standing modes permitted by the cavity). The optical frequency comb will be made up of frequencies separated by $80$ MHz, and will have an amplitude envelope given by that of a single pulse. If the pulsed output of the laser is incident on a photodiode which is then attached to an electrical spectrum analyser, then the modes of the frequency comb can be seen, and will stretch from the $0^{\rm{th}}$ order, with $80$ MHz spacing, up to the limit of the electrical response of the photodiode/electrical spectrum analyser.
I find it best to think of optical diffraction and Fourier transforms when thinking about optical combs in the time and frequency domains like this (the maths is the same and the physics very similar, in my opinion). Think of Young's double slits: Wide slit spacing implies closely spaced diffraction fringes, and vice versa.
Another example of when they arise is during high-harmonic generation when driven by pulses with durations longer than a single optical cycle (denoted $\tau$)**. In this case, EUV can be generated every half-cycle of the laser, giving a frequency spacing of $2/\tau$, (so producing a frequency comb peak at every odd harmonic of the driving field). So, if the frequency of your driving laser is $300$ THz, the frequency spacing will be $600$ THz. Harmonic numbers around the $5000$ mark have been generated using this technique, which allows for frequency conversion from an available (pulsed) laser wavelength into the EUV, where traditional lasers are not available.
2) Enhancement cavities are sometimes used to generate EUV for many reasons. I'll only include a few for brevity:


*

*They allow for the coherent addition of many driving pulses, meaning that low energy, high repetition rate laser sources can be used to generate EUV rather than low repetition rate, high energy sources. This can give you more EUV flux, although coupling the EUV from the cavity can be difficult.

*Generating EUV in an enhancement cavity can mean that EUV bursts can be emitted at MHz repetition rates, where (except for in a few laser systems) the repetition rate is generally restricted to a few tens of kHz.


It is important to note that it is the driving pulse which is circulated and coherently added with each roundtrip of the enhancement cavity, not the EUV. The EUV is generated within the enhancement cavity by the enhanced circulated driving pulse, but is switched out of the cavity after being generated.
The second point in the list above is an attractive one. If you have a given amount of EUV power, reducing the modes increases the amount of power per mode, thus giving a higher signal-to-noise ratio for any frequency comb measurements you would like to make. Also, using the enhancement cavity gives the PHz EUV frequency comb a MHz modulation, which is then used as a more useful frequency reference (see comments below).
EUV isn't always generated using enhancement cavities. There are other methods of generating it using high-harmonic generation, and other means such as x-ray free electron lasers and synchrotrons. The latter two are generally national facilities and beam time is scarce and very expensive, so high-harmonic generation has seen a lot of development recently to increase availability of coherent EUV radiation.
3) If by "phased locked lase[r]" you mean modelocked laser, then yes, that would be an EUV frequency comb. However, EUV is ionizing radiation and is therefore absorbed by pretty much everything, meaning that 'traditional' lasers aren't available for this frequency band. As such, a modelocked EUV laser is hypothetical.
*I think it's great, but I've been told that it isn't a "viable" academic source so many times...
** In the single atom model only. The plasma produced during the HHG process can cause the harmonic spacing to increase which, while maintaining the comb structure, means that it can no longer be used as a measurement reference.
A: The answer and discussion by user113857 and Emilio Pisanty are excellent. But, I wanted to present a more concise answer to the title question:
The generation of an EUV frequency comb does not require an enhancement cavity. However, because frequency combs typically operate with 20+ MHz repetition rates and high-harmonic generation requires high peak intensities (typically >$10^{13} \mathrm{watts/cm^2}$), an enhancement cavity allows for efficient high-harmonic generation with reasonable (sub-kilowatt) average powers from the laser system.
While I couldn't find an example where a true optical frequency comb (pulsed laser with carrier-envelope-offset and repetition-rate stabilization) was generated in the EUV in a single-pass geometry, this 2011 Optics Letter demonstrates single-pass generation of HHG up to the 15th harmonic using a 20 MHz laser. This result demonstrates that (a) it's not completely necessary to use an enhancement cavity to generate an EUV comb but (b) the achievable photon energies for 20+ MHz HHG are much higher when an enhancement cavity is used to increase the peak intensity.
