"If I had more time, I would have written a shorter letter"
I reached the conclusion: Yes, it is possible and easier than you might expect depending on the device. Specifically, to boost a 2.5V-3.0V power supply to power an MCU with radio, it's tricky and it might be a better idea to change the battery configuration. Radio transmission draws more current than low voltage boost converters seem to be capable of providing can easily provide (though with caps in the right places, it seems possible to address this).
Edit: But, thanks to @Jens, I realised that the ESP8266 can actually be powered with as little as 1.8V! So for some devices, you may not need anywhere near the voltage you think you do.
Here's how I reached that conclusion...
Boost module (eBay special)

To start off easy, my first port of call was to use an 'eBay special' boost module. I found one that works with input voltages from 2.5V, which seemed like just enough, and can output from 5V up. My MCU dev board has an efficient 5V regulator, so I figured that'd be fine.
I tried to find a 2.0V/2.5V to 3.3V converter, since that's all the ESP8266 MCU really needs, but it doesn't seem to exist, at least not on eBay. Probably because there's no demand for it, which tells you something. I'm betting there aren't many scenarios where you can't just use the appropriate battery configuration.
Unfortunately, while the scope showed the boost module was outputting 5.04V max, it also showed min 1.92V (average of 4.90V). The MCU seemed to struggle with this unreliable power source and occasionally reset.

The boost module seemed quite happy when the WiFi module is off (the MCU is only drawing 20mA or so), but when you turn WiFi on, the load current demand goes up, and you get quite noisy output on the boost circuit, and it eventually seems to collapse to VIN. Even at higher voltages like 3V, this seems to happen.

Also, when the motor draws power, the battery voltage will cause the voltage to dip below 2.5V if the batteries are coming toward the end of their life. I think perhaps adding a dedicated power wire to the battery just for the MCU could help here, but in any case, it'd be nice to squeeze a bit more power out of the batteries.
Conclusion: It's hard to find a pre-built module that will boost very low voltages to power something like an MCU with a radio.
Edit: Ok, thanks to @TannerSwett I tried something again. I had previously tried to use caps to solve the problem with low voltage dips, with no success. I gave that another try today, in a few places; just after the boost module, just before the MCU. It seems that a 10µF cap near the MCU helps a little, and working up through 100µF, 220µF and 470µF near the MCU, that last one seems to do the trick! Curiously, putting the cap near the output of the boost converter does not solve the problem. What I noticed is that when the MCU goes into deep sleep, that ironically seems to draw a relatively large current for a very brief moment, not sure why. Anyway, the 470µF cap addresses that.
Roll your own boost circuit
For fun, I built 3 boost circuits, one controlled by an MCU, one a 555 timer, and the other by an op amp. All 3 suffered the same fate, which was that they were somewhat inefficient (not ideal for battery powered devices) and had a huge flaw: The MCUs I have need at least 3.3V, so that's out. The 555 timer needs 5V, far too high. And, the LM317 op amp needs 3.0V (but I found it does still work at lower voltages).
Conclusion: It's better to use an IC designed for the purpose which has it's own built in timing circuit that runs at low voltage.
MAX608 boost IC circuit
Curious to see if I could come up with a better boost converter that works at even lower voltages, I built my own boost circuit around the MAX608 IC, which boost voltages from as low as a 1.8V input.
Components:
- Inductor: 22 µH, 950 mA, 0.37ohm 20Mhz
- Transistor: TN0702N3 20V 530 mA MOSFET N
- Current sense resistor: Wire, measured using Mehdi's method (I intend to replace this with an actual current sense resistor)

This had very similar behaviour to the eBay special boost module; my circuit would perform fine at low load currents, but as soon as the WiFi started, the output voltage would collapse to VIN.
I was wondering why higher load currents (well within the advertised 1.5A limit on the 1st page of the datasheet) cause the chip to shut down. Turns out there's a minimum start-up voltage which depends on the load current that might have something to do with the problem.
Edit: I did try using extra caps to solve this and didn't have much luck, but I will likely try this again.

Very important to consider: e.g. a 2V supply/input on the MAX608 (5V output configuration) cannot power much beyond what looks like ~20mA.
Looks like it might not to be possible to drive a current such as 100mA from 2V (at least not with this chip); so powering an ESP8266 (which can draw 170mA with WiFi on) is probably not possible with the MAX608.
Edit: Actually, now that I'm having better luck with the 'eBay special' boost module, and also based on the comments, I think that the start-up current might not be an issue.
Different battery configuration
The first idea I actually implemented was to replace the two C batteries with a single 9V, and use a regulator, but the L78L quiescent current (2mA) drained the battery faster than I'd like. 9V alkaline batteries typically have only 550 mAh, compared to alkaline C's which have 8000 mAh. I also needed 3V to drive the motor, and that required a 2nd regulator, more quiescent current to drain the battery.
To get back the benefit of the longer life C batteries, I also tried adding 2 extra AA batteries (in a different part of the device) in series with the C batteries to boost the voltage to 5V-6V. Not sure if that's particularly safe (perhaps that's a separate question), but it worked well enough. However, adding batteries was a real hack because there was just enough space to tape a couple of AA's together and solder some wires to them (a bit tricky to replace when they eventually expire).
Seems like using 4 AA batteries is the simplest way to go, but that'd require me to modify the battery compartment which is fine, but I'm curious to find out if there is a boost converter circuit that would work before I get the power tools out. Also, I still need a 3V power source for the motor, easy enough to do by taking power from half way through the battery series, but adds a little bit of complexity to the physical case modifications.
Conclusion: Perhaps don't try to boost a 2.5V-3.0V power supply to power an MCU with radio; use 4 AA batteries instead.
Just connect the ESP to 2.5V
"The ESP8266 should run with 2.5V supply, probably you don't need a boost converter anyway, just a good ceramic or tantal capacitor."
Edit: Thanks @Jens! And after all that, I connected the 2.5V supply to the 3V3 pin and it worked fine. Actually no capacitor needed, and incredibly, it even works as low as 1.8V! This is far beyond what the ESP8266 datasheet specifies. It even seems to establish a successful WiFi connection and successfully send and receive data at such a low voltage. I was obviously looking for an excuse to play with inductors.
Conclusion: See how little voltage you can get away with when powering an MCU.