Usually, there's no reason to embed more than more one profile. With an input image, you embed the input profile, so that your colour management system can work out what colours the numbers in the input data means.

No reason to embed a monitor or output profile - they're nothing to do with the image data, and if you move the image to a different system, you'd need to use a different monitor profile for viewing, or a different output profile for printing to a different printer or paper type.

Normally, the monitor profile is associated with the system you're viewing the image on - and you'd use that profile for all images you wanted to view on that monitor. (The monitor profile lets the colour management software figure out what values to send to the monitor to get a particular colour).

Similarly, the output profile lets the colour management software work out what ink values to send to the printer to get a particular colour. That varies with printer, ink, media and printer settings. Again, this is usually set by the printing software. If you're producing something like a CMYK file for conventional printing, then you might embed the output profile in the CMYK image. In this case, the colour management software could use the reverse cubes in profile to work out what colour the CMYK values in this file mean (for example, if you wanted to output the file on a different output device, where you'd need to go originalCMYK->colours->newCMYK using a different output profile).