For all intents and purposes, Japan never really fought coordinated campaigns with Germany and Italy. Italy was essentially out of the war by 1943.

However, along with these three "main" axis powers were several other smaller players.

Puppet states such as Vichy France and Manchukuo activity fought for the Axis.

Romania, Hungary, and Bulgaria assisted Germany until they switched sides in the 1944-45 Russia counteroffensives.

Finland, while not closely aligned with the Axis powers, did coordinate with Germany to win back lands they lost in the Winter War.

Edit: Hungary never switched sides, it attempted to exit the war and was subsequently invaded by Germany. Credit to /u/TheLegitimist.

The short answer is sort of. It really depends on how one defines the Axis alliance and what it meant to be an ally within that framework. It is important to understand that Germany, Japan, and Italy were often allies in name only and had disparate war goals that often created problems for each other; especially with regards to Italy and Germany.

Italy simply wasn't ready to engage in modern warfare and found itself outclassed in almost every engagement it fought in, (the possible sole exception being its Navy.) Furthermore, the differing national goals of Italy (North African and Balkan expansion) and Germany (eastward expansion into the USSR) meant that both nations' forces were diverted to deal with military situations that ultimately sapped power from those nations' primary war goals. Germany's deployment of forces to the Balkans, Greece, and North Africa meant that there were that many fewer troops able to fight on the Eastern Front. Conversely, Italy found itself committing troops that were ill prepared and equipped to fighting in Russia. The German and Italian partnership was never an equal one and Italy found itself getting the short end of the stick frequently.

Regarding Japan it can be said that their membership in the Axis pact was really in name only. Besides being on the other side of the world Japan had war goals that differed greatly from Germany. Outside of some isolated military engagements and limited technological cooperation there wasn't really any direct benefit to the alliance. If anything the ability of Japan to assist the others was largely tangential. The Japanese war against British, American, (French, Dutch,) and Commonwealth troops tied down forces that could have been used to fight Germany. Beyond that, Japan's refusal to fight the Soviet Union in Outer Mongolia and Siberia meant that Stalin was able to divert divisions west to fight the German invasion. Had Japan chose to strike north and engage the Soviet Union it is very possible things would have been different regarding the Eastern Front.

So where does that leave the alliance? Well, there were a number of "vassal" states that committed to fighting alongside Germany. Bulgaria, Romania, and Hungary threw in with Germany and committed troops to the invasion of the Soviet Union. They were poorly equipped and led and were most notable for being the achilles heel that allowed Soviet Forces to encircle the 6th army at Stalingrad. Finally, Germany, Japan, and Italy had client states in the form of occupied territories and colonies where manpower and materiel were drawn for their war effort.

Finland was a de facto ally following their Winter War with the Soviets. While never openly stated by the Germans or the Finns, Finland received aid from Germany and prosecuted their own war against the USSR in parallel with the Germans. Frustratingly for the Germans the Finns never cooperated in concert with the German military with one such result being the ability of Leningrad to withstand siege, mainly by the Finnish refusal to seal off Lake Ladoga.

Likely the closest you'll find to the Axis powers inviting another country to join them as an ally is Franco's Spain. Hitler actively lobbied Franco to get Spain to join the Axis powers. However, Spain had just recently concluded a costly civil war and was in no position to enter a new conflict. Franco knew this and actively demurred on committing his nation to war. This annoyed Hitler given Germany's contribution to Franco during the civil war. However, Spain did commit volunteer groups ,notably the Azul Division which fought on the eastern front, and provided clandestine support for German activities.

tl;dr: Spain fits the bill but the axis powers were complicated and there were more than four members in reality if not on paper.

Source: Antony Beevor's excellent The Second World War

Note: I am away from my sources for an extended period of time and other then Beevor don't have access to more. I am sure I am missing some things and glossing over others and look forward to other's filling in, correcting, and expanding.