How many port adapters, and which ones can be stacked up and actually work?

None of them will do anything you expect. At all.

The green one is a USB to PS/2 converter, it is a wire converter rather than signal converter. It will only work if the device you plug into it can detect how it is connected and speak both languages. Your USB stick will not do that, a USB mouse might.

The next is a PS/2 mouse to serial converter, again the PS/2 device needs to understand both connections. A USB stick cannot talk through this.

Next is a 9-to-25 pin serial adaptor, nothing special, just more unconnected wires. No talking from USB.

Next is a 25 pin gender changer, allowing the 25-pin female serial converter to plug into a 25-pin female parallel port. Neither the parallel port nor USB device will understand the meaning of whatever wires are wiggling at them, if any of them are at all.

If I get bored later I might try to chase the theoretical pinouts, but it seems a bit pointless in all honesty.

It is a useless mash of connectors that will achieve nothing.

To answer the "at what point does it stop working" part, it depends what is being converted.

I personally have an old AT keyboard, into a AT/PS2 converter, which connects to a PS2/USB converter and connects into a USB KVM. That's two adapters in a row, or three if you count the KVM.

Any number of extenders or joiners would work, up to the point where losses across joints or total cable length is excessive. So 100 1 metre ethernet cables and 99 RJ45 joiners might work, but that's not an adapter and is against the spirit of your question.

I believe the picture shows a general RS-232 full-size DB25 COM port, and not a parallel port. Technically the stack-up should work with a special USB/PS/2 compatible mouse, which operates in LS USB mode (1.5Mbit/s).

However, the picture shows a USB stick, which can operate only at FS rates (12Mbits/s) and above. This "setup" will not work because the setup can operate only at 1.5Mbit/s USB 1.0 rate, while the FS USB device needs serious processing of data signals at 12 Mbit/s, and needs a carefully scheduled special service from PC host, which cannot be provided by COM port.

So the simple answer is: the COM port PS/2 to USB converter cannot provide the necessary communication speed nor proper USB protocol for a FS/HS memory stick. Specifically it "stops working" between the stick and green USB-PS/2 adapter.

ADDENDUM1: one fundamental evidence that this setup is a joke is that neither EPP or COM ports have any 5V power, which is necessary to power the USB stick.

ADDENDUM2: yes, this is the PC parallel port, per description of DELL 2550 sever, and thanks to "plugwash". The PP is worse, since PP does not have any UART serdes conversion hardware, and bit-banging of the port from x86 PC is clearly out of range for 12Mbps receiver processing (which needs 20ns sampling/reading rate).