100 ohm would give a max current of approx 5mA (FV 2.5v off a 3v supply is just 0.5v across the resistor),
as long as you don't have the leds connected when on 5v, you could come down to 47 ohm to give 10mA.
680 ohm as suggested above is pointless if you want to be able to see the leds in daylight.
Neil.
edit: have the leds lit with the pic pin low (LED anode to Vcc) as the pic can sink more current than source (20mA max)
Very true, as had been mentioned in many threads prior. For a 20M2 Picaxe at 5 volts, a pin sourcing current already includes about 90 ohms of resistance. The same pin sinking current already includes about 30 ohms of resistance. These resistances are device and voltage dependant.
At a supply voltage of 3.0, one would likely need some small additional series resistance if sinking current through a red LED. Conversely, a blue LED driven by the same pin sourcing current may not light at all.
The devil is in the worst case numbers, and how safe you want to be. In Hippy's example, the LED's are between pins sourcing current through a (probably) 90 ohm resistor, and then sinking the current through a 30 ohm PWM resistor. There is already about 120 ohms resistance in series with the LED, and then PWM on top of that. LED's of all types can withstand, repetitively, very large currents of very short duration. That is how the LED flash on your smart phone works. One amp for a microsecond is one microamp average, but it will be VERY bright for that microsecond. As long as you don't exceed the AVERAGE current limits, you can even do it over and over again.
You really need to run the worst case numbers for the specific condition. Highest possible supply voltage, lowest possible LED forward voltage, lowest possible intrinsic pin impedance, probability of device survival as currents approach, or exceed, maximum permissable levels. Then make an informed decision based on what you know. If the LED is going to have to flash only once, ever, then you can use the most efficient red LED you can find (lowest possible forward voltage with the greatest light output), connect it between the 3 volt supply and a low pickaxe pin, and expect it to reliably flash very brightly. Exactly once. It (or the Picaxe) may actually survive several, or even millions, of such abusive cycles, but by analysis you know that they are not obliged to do so. But you CAN expect it work once.
Or you can take what you know, and add resistors to the point that the probability of failure approaches that of the unstressed lifetime of the most failure prone device (probably the picaxe, since it is the most complex). In that case, the circuit will be switching current as required, and as programmed, but the LED will be so dim as to be pointless.
Somewhere in between those two extremes is reality. Such is engineering.