Is it possible to link multiple computers together to use both their processor power?

Yes.

What you describe is a cluster. I don’t believe there is a practical way to do that with a laptop. You would have to remove the guts and put it in a case with the desktop. The GPUs would not be combined.

But you are thinking along the right lines for high-powered computing.

For an efficient cluster, you combine a bunch of identical motherboards ( picture )

The fastest supercomputers in the world work like that. And companies like Google and Amazon run their cloud services on thousands of clusters.

This guy did it already so I think it probably is possible :D

The key phrase is “Beowulf cluster”, and odds are that you will be running Linux of you build one. BTW, the fastest supercomputers are clusters, and most run Linux.

There are apps that let allow unused CPU cycles to get applied to some problem like protein folding or SETI detection. Takes some setting up on the host end especially to divide the work into bits that can be delegated.
I haven’t heard of any clustering system that accomodates GPU sharing though, only CPUs.

@dabbler They exist. GPUs are powerful enough to be considered computers in their own right, and some people do take advantage of that.

@jerv Certainly folks are using GPUs for some serious computation, including some of the most productive BitCoin miners, and probably some market simulation engines.
I haven’t seen anything about them being used in clusters. But if you have the I/O to exploit clustered GPUs why not?