What is the newest version of Photoshop compatible with this older laptop?

The Intel HD graphics is the most obvious shortcoming here. A more powerful GPU has really been a basic requirement for a very long time. It may work and it may not work, but either way it's borderline.

The oldest version available is 22.2. This very old version has been kept alive because it was the last version with working 3D. Otherwise the policy is the current major version and the one previous. So in the CC desktop app you will find version 26 and version 25 - and 22.2.

If you come across anything older offered on the internet - don't even think about it! It will be pirated and illegal, not as a probability, but as a dead certainty.