"Parallel" 74hc595 possible ?

A normal 595 can handle about 70mA supply current max according to the specs (nxp).
The tpic6b595 can handle up to 500mA, but costs 10 times as much.

I'm not looking for other chips which may as well be available, but was wondering about the next.

Suppose I tried to increase the max current by soldering an identical 595 on top of another 595, would that work or... would it be a bad idea ?

no electrical engineer, but I think it could work.

Give them a separate power supply too? GND connected
// 8 * 70 = almost 0.6A 16 * 20 = 1.2 Amps

In general it would work. However if for any reason they got out of sync with each other they would destroy each other.
It is best to use the 595 to drive another chip like a UNL2003 or similar device, it's much cheaper.

A normal 595 can handle about 70mA supply current max according to the specs (nxp).

I think you are not understanding this, it's not the maximum current for a pin.

The solution will be unstable and unpredictable and may even shortcircuit if the timing is "right".

Like Grumpy_Mike suggested, invest in components that do what they are designed for, to make a reproducable and stable project.

Cheers

70mA max is total supply current for the chip as far as I understand.
Individual pins can handle about 20 mA max, average would be about 70mA/8 when all are driven.
I could of course limit a led to about 8.5 ma.
But I'm using POV on a 5x5x5 cube I'm building and power one layer at the time.
8.5 mA for 1/5th of the time might be little. ~60mA per pin (tpic) is more then enough, but limiting 'm to 16-17 mA (stacking 2 standard 595's) or ~25mA (using 3) would probably be enough.
3 would ofcourse also make the possibility of wrong timing worse.
Getting out of sync is a point I was wondering about, one pin showing 5volts and the same pin on the other chip having showing 0 volt doesn't sound promising.
By buying identical chips, (preferably from the same batch) I hoped to eliminate that possibility as much as possible.
Point is, I never tried it before, don't have too much knowledge and was wondering whether others have tried it successfully.
Btw, the cube's ready, the rest of the circuit is still on the breadboard.

Hmmm, apparently I've spent far too much on my previous ULNs, just looked up some prices, is indeed a better option to prevent mayhem and not as expensive as the tpic.

Thank you all !