Usually it would not be the best practice because of variable leakage, but this kata is made for this xD.

Fails for deepCompare(null, {}), same reason as @bbroeksema points out. typeof null === 'object'. I was trying to parse the way you return on === and on !== 'object' to see if there was some clever avoidance, but it's not so.

I haven't found a way to avoid doing a null check at the top and a === value compare at the bottom, separately.

There is (unless it was just added recently), but it had null as o1 and undefined as o2, so the third line of your code (typeof o2 !== 'object') returned false.

Certainly true.

Then there should be a test for that, right? :-)

This doesn't work for:

deepCompare({name: 'blah'}, null);

typeoff null returns 'object', so it will pass the third line, next Object.keys(null) will throw an error.