Let me sum up your limitations of a CR2032:
- 10mA is about the max current you
want to pull from a single, it is
easy to put them in parallel, but a
large amount of testing(more than
2000 batteries worth) has confirmed
this.*
- They can be purchased to have
400mAh, the less current you pull
the closer to this it will be,
pulling more then 1mA decreases this a decent
bit.**
- Under a 1mA load they will
decay all the way to 1.5V before
they fail, they will be at 2.7V
almost right away.
- You can measure an almost full voltage on them with a multimeter when they are dead. This is solved by placing a load on them.***
- If you are lazy, it is very easy to tell how much charge they have left by how much they make your tongue tingle. Your tongue acts as the load and measures. This is probably by far the easiest way to test them, although it does pull a decent bit of current.
I think Thomas wrote a good answer, I just thought it might be helpful to give some details of the coin cells since it seems you have used AA quite a bit.
*Wikipedia says up to 15mA pulsed, but we confirmed that up to 1mA shows a nearly consistent capacity.
**Wikipedia shows a standard that is a bit lower, but my company would always purchase 400mAh or 450mAh CR2032. When you buy a "standard battery" you can expect 200mAh it seems.
*** People often will measure batteries without load, when someone tells you on a project that ran out of power early, ensure their original battery measurement was under load, very easy mistake.
Oli gave a good answer, but wait, mine will be better! :-)
Oli commented on the limited current from the coin cell, and that's indeed something to keep an eye on. This CR2430 cell gives 5 mA as maximum continuous. Let's see if we can manage that.
It's a good thing that you only need one LED on at a time, otherwise I would even consider the coin cell. This looks like a nice LED: typically 15 mcd at 2 mA.
Oli went for a SIPO (Serial-In, Parallel-Out) shift register for the LEDs and a PISO (Parallel-In, Serial Out) for the buttons. That saves you a lot of I/O but costs extra components. Can't we use the I/O of a microcontroller directly? 22 LEDs and 8 buttons is 30 I/Os, no problem, but we can do it a bit cheaper if we multiplex the LEDs in a 4 x 5 matrix. Normally this would decrease the LEDs luminosity by 75 %, but since we only have to light one LED at a time we can select one row and one column statically. So we need 4 + 5 + 8 = 17 I/Os.
Usual suspects for a microcontroller are Atmel AVR and Microchop PIC. Usually I'd avoid PIC for LED driving because it can't source or sink 20 mA, but we have a low LED current so no problem. PIC is also cheaper than AVR. The PIC16F57 has 20 I/Os, so that's enough. The datasheet says 22.5 µA maximum for a 32.768 kHz clock at 2 V, so at 3 V that still will be below 50 µA.
That's it. A microcontroller, a cheap crystal, 22 LEDs, 8 buttons, and 12 resistors (4 for the LEDs and 8 for the buttons. The PIC16F57 doesn't seem to have internal pull-ups). No shift registers needed.
Best Answer
From a voltage standpoint, yes it could be done.
However, voltage is not everything. You need to look at the datasheet for the coin cells to see if they can supply enough current. In almost all cases (apart from some Li-Ion rechargeable ones), the rated continuous current sourcing capability is only ~2mA at most - typically only 1mA. Though it should be possible to run at a higher current at the expense of capacity - but probably not more than 10mA or so at a guess, depending on the battery.
With four in parallel, you can probably only supply 8mA to the board without significant voltage drop. The downside of running them in parallel is that if the batteries have slightly different terminal voltages, the lower voltage ones start acting as a load rather than a source).
By comparison, an AAA battery can typically source as much as 1A continuously, and also have much higher capacity.
Coin cells are designed for battery backup, or ultra-low power applications. They are not designed as a replacement for AAA batteries.
Having said all that, it is hard to tell what the requirements of the micro:bit are - in a quick look around I can't seem to find any information on power supply requirements/current draw. However looking at the processor on it, that requires at most around 11mA which may be within the capabilities of the coin cells. I'm not sure what else is on the board, however if you start trying to drive the LEDs, that will increase the current consumption far beyond the limits of the coin cells.
As a final note, you are getting your units mixed up - current and capacity are different things:
Current is a measure of the flow of electrons, with units of Amps [A] (mA = milliamp = 0.001 Amp).
Capacity is a measure of energy stored in the battery, and is measured in Joules, although frequently quoted in Amp-Hours [Ah] (mAh = milliamp-hours).