Cheap voice compressors that do not take up too much space
Ready-built
ART Tube MP/C
Looks cool. ~€110. This one's possibly made for guitars, not sure. No headphones output.
Behringer Xenyx Q502USB
This is a mixer, and audio interface. But it has a compressor and tone controls and it's the cheapest on this list by far! €45. Headphones output!
Rolls CL 151
On-line reviews seem to indicate that people either hate this one or tolerate it. It seems to have a non-standard behaviour. ~€130
No headphones output.
Amazon.com: Peavey HB2 Headphone Amplifier
This looks great but I think it's discontinued. The HB2 is specifically made for headphones listenting, I guess for a live band member so that he can hear the others clearly without suffering the ill effects of sudden bursts. This is the closest to my use case.
Semi-built
SSM 2167 pre-built
This one will give you 30dB (I think) of gain that you maybe didn't want, and I am unsure of how to desolder or switch in components for changing compression and limiting.
Teensy audio
The audio card for the Teensy SOC. The audio chip has built in compression possibilities, so can be switched on and configured in two lines of code. ~€20 + ~€20 for the Teensy + cabinet and knobs. This is a powerful solution with I guess close to instaneous boot times. But it does mean you have to do 28 solders, and use the arduino IDE with at least a few lines of C code. Having done that though, you can deploy different kinds of code.
Raspberry Pi + Jack + Rakarrack or similar
A Raspberry Pi with JACK and a USB audio interface ~€50 + cabinet and (possibly USB) knobs. This is the easiest semi-built option by far, since it requires no soldering or mounting of other components. JACK can patch any sound input or ouput to and from any JACK-compatible software. When JACK is installed you can use different plugin hosts to drag-and-drop a sound processing chain from your sound card's input to its output. One such plugin host is Rakarrack. Remember to use a low latency Linux kernel and to prioritize audio.
The more I think of it, the more I like this solution: You can prototype on your normal laptop or desktop Linux machine, there is a plethora of ready-made effects of which some are of high quality such as in my experience the Calf compressor. And there are hundreds of more plugins.
The things that I am wary of is how quickly it would boot and how stable it will be in practice. Also, although it's no harder than any other option in this section, what kind of user interface you will want to have. You could go with a touch screen, with potentiometers directly wired to the Pi's inputs or with a USB controller. I'm gonna do this one! At least prototype it on a laptop.
Roll your own
SSM2166SZ Microphone Amplifier IC SO-14 Analog Devices
SSM2166 - more configurable than the previously mentioned 2167. It has a a limiter and compressor built in that are easily configurable. But there are no breakout boards for the 2166 afaict, and it only comes as a surface mount component. Although of the soldering-is-humanly-possible surface mount component kind.
Conclusion
Well, all the above options exist for a reason, but I'd recommend the Behringer or the Raspberry. Unless you are a serious tech-head and might think of selling your work to others.