What size wattage audio speakers should you use for your 330 amp Harman-Kardon audio receiver?

First, determine the true output of the receiver. By 330W do you mean 165W per side stereo? Or is that the sum total of the 5 amplifiers for a surround sound system? Or (and this is very common) are you just going by the 330W figure on the max power consumption on the back of the receiver? Max power consuption of 330W typically means you've got ~200W of power split between however many amps. Look it up in the manual, go online if you have to. HK will have the true specs available somewhere.

The speaker's RMS rating should match or exceed the rated output of the receiver to be safe. If they're efficient, i.e. 90dB@1W or better, you can get a way with a little less, but it all depends on whether the people turning the volume knob know what overload distortion sounds like.

I should also point out that if it's a certain volume you're looking for, i.e. "as much as possible", then you should really have selected speakers first, then found the right receiver for them. For example I have 75W Technics Linear Phase speakers from the 70's that will crack teeth at about 40W because they're stupidly efficient, but not the sweetest sounding speaker.

Like they say, you can have high efficiency, great sound, and a low pricetag - pick two of the three. The balance of those three speaker aspects is probably the most important decision when choosing audio equipment, from a bedroom mini-system to a stadium line-array.