Ok, just my two cents worth..
Most powerful WW2 Axis possible includes Spain, Turkey and Sweden.
Two points of diversion I can see off the top of my head.
The Axis delays Barbarossa and throws everything they have for a knock-out in the Med against the U.K.(without declaring war on the U.S.). IF (and thats a big IF) the Axis can drive the Royal Navy out of the Med (taking Gibraltar, Egypt and Malta), the British hold on the Med will wither on the vine. If that happens, the U.K. is going to start to feel the pinch. Would it bring down Churchill? Not sure..
The Axis still goes forward with Barbarossa. Some differences with the northern sectors. In OTL, the Finns did take part of the Murman railway (for a short time). IF the Swedes and the Finns go balls-to-the-wall during Barbarossa, Murmansk and Leningrad fall, almost without a doubt. Will Arkhangelsk fall? Maybe/maybe not. They might be cut off from the rest of the USSR. What effect will this have on the war? I'd love to hear some opinions.
You could actually have Malta fall fairly easily without adding any new Axis nations. Have Mussolini attack Malta from the word go instead of invading France. France was going to fall anyway and it wouldn't have been that hard for Mussolini to get Hitler to give him some border territories as part of the spoils.
Although the Italian military was not very good, Malta was under-defended at this point in the war, and should Italy attack they would have no way of being reinforced or resupplied, not to mention that the Italians would have total air superiority. This happens, Malta falls in June. You also have Spain join the war, then Gibraltar falls not too much later, leaving Egypt terribly hard to supply. Have Italy leave Greece alone and focus his attacks on Egypt, the extra Italian attention, the addition of Spanish troops and the difficulty of supplying the defenders make Egypt a lot easier to take, also the pro-Axis uprising in Iraq is not put down so easily.
With the Med fast becoming an Axis lake, Turkey and perhaps even pro-Axis Greece join the Axis. When Babarossa comes around (Perhaps sooner without all the fighting in the Balkans), Turkish and Greek troops surge up the Caucasus, taking Baku and perhaps some other important cities. If you have the Swedes joining, and them and the Fins going full-out, Leningrad and Murmansk fall and Arkhangelsk is put under siege if not taken. If you have Hitler allow Lithuania to be a semi-independent puppet state, much in the manner of Slovakia, that gives the Axis a boost too. This coupled with the earlier assault and the addition of Spanish troops, Moscow is now under a much more serious threat. Will it fall? Perhaps, i have no idea, but the USSR is in a much weaker position.
Another thing you could do is have the Yugoslavian coup fail, keeping them in the Axis. It's also possible to have Italy pressure San Marino into joining the Axis, this will make almost no difference, but you could do it.
With Spain in the Axis, i don't see it as impossible that Portugal would send a volunteer division to the Eastern front, much in the way that Spain did IOTL.
For one last blow to the Allies, remove Hitler's indecision and have the Germans run right up to the channel in the Fall of France, capturing or killing the BEF, making Egypt an even softer target.
I could also return to a point i made earlier about a British invasion of Norway. A good way to go about this is to have the British invade only hours before the Germans were planning to, this way the Germans will still invade Denmark, and will be fully ready to resist the invasion of Norway and placing Norway in the Axis. Another benefit to a British attack on Norway is that it makes Sweden that little bit less friendly towards the Allies, and therefore that little bit more likely to join the Axis. Norwegian troops will likely be primarily interested in self-preservation, sitting around in defensive positions, but they'd no doubt make some effort in the invasion of the USSR, putting Arkhangelsk under even more threat.
The other potential Allied invasion of a neutral country is Afghanistan. In October 1941 the Soviets were going to if an ultimatum wasn't agreed to. We can assume that the Afghans say no, and the Soviet invade. They'll last a month, tops, yet will have a resistance movement of sorts (Look at any other time in modern history Afghanistan was invaded), and it's just another drain to the Soviets.
So without bringing Latin America into it, or having a pre-1939 POD, this is the best i can do. Not sure what outcome this will have but with Norway, Sweden, Greece, Spain, San Marino, Turkey, Lithuania and Afghanistan now in the Axis, and Iraq and Yugoslavia kept there longer, plus with Malta and Gibraltar both in Axis hands by the end of July 1940, the Allies are in a perilous position.
Final thought: Mussolini forgot to warn his merchant fleet about the declaration of war IOTL, causing 2/3 of them to be seized by the Allies in the first week of war. Have him warn them for a slight boost to the Axis.
