Looking at which European powers* were still neutral (and uninvaded) for most of WW2, there's not a long list: Spain, Turkey**, Switzerland, Sweden, Portugal, and Ireland.
Ireland, Portugal and Switzerland can be safely ruled out, I think, barring divergences so far back that WW2 as we know it is unlikely. Getting Spain to actively join the Axis would have taken some doing, from what I understand (infrastructure wrecked by civil war, extremely vulnerable to Allied attacks). Turkey would be a struggle to get onto the Axis side as long as İnönü was president***, but I'm not familiar enough with Turkish politics of the era to know whether a different government in Turkey would be any more likely to join the Axis.
Sweden had a long tradition of neutrality by WW2. Getting it to change that would take some doing. Getting it on the Axis side would be even harder. Possibly if the French and British went ahead with a serious plan for an intervention in Sweden to capture the iron mines or to force intervention during the Winter War, you might get a Sweden which is at war with those countries for a while, but that would be unlikely to turn into serious Axis membership.
Outside of Europe, there's not many plausible candidates who didn't already join one side or the other. Iran is perhaps the most plausible candidate to join the Axis (though still not very likely), but given what happened to them in OTL with the Anglo-Soviet invasion, I doubt that Iran would be an Axis power for very long even if it joined.
* Microstates such as Andorra are not "powers."
** Yes, I know Turkey entered WW2 with a few months to go.
*** Well, unless the war was going so well for the Axis that they appeared to have an easy victory, which seems unlikely.