changing 3 watts speaker to 30 watts

You are clipping your amp. Doesn't matter what the speaker is rated at. You can't turn it up to max volume. If you need to go that loud but something else.

This is like that movie Frequency, but with a computer instead of a Ham Radio... is there Aurora Borealis going on? He's posting from 1969, right?

Sure, you can upgrade your speaker, assuming you find another speaker driver that fits the physical dimensions of the existing location, but you're not going to escape the amp clipping.

It could be possible to upgrade both the driver and the amp, but now you're talking about replacing everything, so you might as well dump whatever that's not working for you and buy a better item in question.

If you just want a speaker that won't break when you put the volume knob to max, perhaps try a 6 or 8 ohm speaker. That should put less load on the amp, perhaps preventing it from clipping. But it probably won't play as loud, perhaps by a significant amount. You could also mod the circuit a bit so that "max volume" is not as loud, protecting the speaker.

If your requirement is that you want it to be as loud as possible with the internal amp, find a speaker with the highest sensitivity you can find.

However, I think the other responders are probably right that you will not achieve satisfactory results without an external amplifier. We can continue if you clarify your requirements -- why are you putting the volume knob to max?

-Max

Believe it or not, on just about anything with a volume knob, you reach the amplifier's max power when the knob is set at 50 or 60% (12:00 or 1:00).