Hi,
It may depend which PICaxe you want to use (the 08M2 is more restricted) and whether you want to use the SOUND command, or just a more general "Beep" or TUNE, etc.. But for a good range of amplitude (Volume) and frequency (Pitch) you probably need to use some degree of "external hardware", but it could be quite simple. For example the DAC output level being "chopped" (modulated) by a diode to another pin carrying the Sound square-wave. Or as Pete suggests, using different pins to select the output drive and/or amplifier gain (via different resistor values).
Personally, I would explore the possibility of PWM-type commands driving a Resistor-Capacitor Low-Pass filter (R from the output pin, C to ground shunting the higher harmonics). The filter will make the sound "smoother" (i.e. with less harmonics) and narrow PWM pulses (i.e. with a low duty cycle) will sound quieter. Theoretically the PWMOUT command even gives the possibility of playing "Chords", but it does require the program-writer to issue a "Stop" (OFF) command at the end of the note(s), after a suitable PAUSE. With a Low Pass Filter, the PWMOUT command can select a Pin, the Frequency (PERIOD and PWMDIVn) and Volume (DUTY).
Other "Low Hardware" possibilities are to use the internal "Digital Signal Modulator" (DSM) in M2s and/or hanging a capacitor on a "Touch" input pin (to generate a "Triangle wave" with various amplitudes and frequencies).
Cheers, Alan.