When using the cleaning cycle on an inkjet printer, where does the ink go?

It's actually called a spittoon, and the printer counts how much ink has been dumped into it just as it counts how much has been put on the paper. If it gets nearly full, the printer will complain, and if it becomes full, it will cease to operate. Cheap printers do not have replaceable spittoons; they're big enough in theory that the printer will be garbage by the time it is full. In bigger printers that are designed to have litres of ink run through them, spittoons are service swappable parts, with degree of difficulty of the swap varying from "unscrew this hatch" to "call a service guy".

In my experience (tho not with that particular printer) there's a sponge that soaks it up, same as when it initialized after having been parked/off.

For my big Epson, it has spittoons about the size of a Trivial Pursuit deck, one each side. For odd reasons, it only ever spits into the right one, but (in true consumables hell) has chips on the spittoons that register them both full if one gets full. That means every month or so you have to swap them over so you at least get some wear out of both of them. A used spittoon is noticeably heavier; then again, this printer does use 750ml cartridges, and sounds vaguely like a jet taking off when it's in full flow.

Reiterating attercoppe: I haven't dug into that printer, or even the particular HP on my desk too heavily, but I was involved in developing the hardware for a printer two years ago (didn't make it to market, got used as leverage for making a deal with a bigger printer manufacturer). We had a big sponge and reservoir on the opposite side of the carriage from the head parking area ('cause the head parking area put caps on the print heads). It was big enough that if you ever managed to fill that up you were going to be our bestest friend in the whole wide world 'cause it would mean you'd bought enough ink from my client that they could make substantial inroads into the national debt.

Yes, like scruss says, our big Epson will quit working when it's time to replace that part. My boss refers to it as the "drool bucket," which I find oddly hilarious.

fwiw, a lot of ink winds up there. lots. more than i would have expected.

My Canon printer's spittoon got full. I had to use some service procedure to reset the count in the printer (pressing and holding random buttons at random, it seemed like, without any feedback) and then used paper towels and cotton swabs to soak up the sponge and it was as good as new.