There are several major European states which remained neutral for at least most of the war and were not conquered or allied with Nazi Germany during World War II - Spain, Sweden, Switzerland, Portugal, Ireland, and Turkey (which we will count as European since they to have a toehold at least). Answering for all of them is too much of a task for me alone, but I've actually addressed Switzerland in the past, so will do a brief recap here, and you can follow the link to the full answer. If you have follow ups though, please ask.

Regarding Switzerland (previous answer, in the early spring of 1940, they did fear the Germans might invade, and there were minor feints to try and make the French think it might happen too, but nothing came of it. The Swiss plateau may offer a path around the Maginot Line, but Belgium was just easier and also more logical. Although plans existed - Operation Tannenbaum - for the possible invasion, it wasn't launched for several reasons. The Swiss would be quick to trumpet their preparations for war as being a deterrent - the famed citizen-militia and the "impregnable" National Redoubt - that was only part of it at best. The simple fact is, why would Germany invade Switzerland when the Swiss were cooperative enough to make it not worth their time? Swiss business mostly was only able to trade with Nazi controlled Europe, Swiss banks handled Germany's ill-gotten gold and boatloads of money as well. Even if Germany was successful in invading, it would be at the expense of commerce and infrastructure that was already at their disposal. Why see the Swiss destroy it all (if invaded, the Swiss would have blown every bridge and tunnel they could while retreating, several of which were very important for the Germans), and lose their industry, when you can basically have it without invading?

Gregory answered the Swiss quite nicely so I thought I would take the Turks. Turkey would likely have been a real drain on Wehrmacht forces as the the forces of the new republic had a fierce reputation as defensive fighters presenting a nasty challenge when combined with the arid plains and mountainous terrain of the country. Perhaps most importantly, especially if they factored in a lot of resistance and guerilla fighting is the fact the Nazis bought many of their heavy metals from the Turks.

Flipping the question on its head a bit, the Third Reich actually had very little respect for the rights of European neutrals. Denmark, Norway, Belgium, Luxemburg, the Netherlands, and Yugoslavia were all technically neutral countries, albeit ones with various degrees of sympathy to the Allies, and even though Greece was fighting Germany's ally Italy, but Metaxas actually tried to get Germany to act as a mediator. If one wants to stretch the definition of neutral, technically the Soviet Union was a neutral party to the war until Barbarossa; its expansion into the Baltic and portions of Eastern Europe were sanctioned by the secret protocols of the Molotov-Ribbentrop Pact and it was Germany that violated the pact, not Stalin. Again, this is stretching the meaning of neutral, but technically, the USSR was a non-belligerent and maintained diplomatic relations with the major waring parties of both sides between 1939-41. Hitler and his military planners perceived that an invasion each of these neutrals suited German strategic interests and had no compunction about violating the standards of international law and diplomacy. This leaves the question of the major European states that remained neutral and unoccupied for the majority of the war. Ignoring the interesting cases of microstates like Liechtenstein and Andorra and states outside the immediate geographic reach of German power (Ireland and Portugal), this leaves Spain, Switzerland, Turkey, and Sweden. The reason for neither occupying nor actively courting these neutrals to join the Axis had much to do with the same strategic calculus that impelled Germany to invade and occupy their less fortunate counterparts.

This strategic calculus enmeshed three interrelated elements: the desire for natural resources/finished goods, overall timing, and Germany's vision of a German-dominated postwar Europe. of these three, resources were the most salient and apparent. Of the big four neutrals (Spain, Switzerland, Sweden, and Turkey), each provided the Germans with a wide variety of raw materials and finished goods like ball bearings that were in short supply in the Reich. Although the popular image of the Third Reich was that of a military superpower, in reality, German rearmament and its war effort was plagued by persistent resource bottlenecks that placed a ceiling on the upper limits of German expansion of its war industries. For example, tungsten was an essential mineral for both high-end machine tools and anti-tank ammunition and Germany possessed few indigenous sources of this mineral. Similarly, chromium was also vital for arms production, and likewise Germany had few sources of it. The omnipresent blockade of the Royal Navy meant Germany could not import these minerals from outside of Europe and thus had to turn to countries that possessed them like Spain and Turkey. In the German wartime economy, the Iberian peninsula provided its tungsten, Sweden high-quality iron ore, and Turkey chromium, and Switzerland precision instruments and other finished goods. The acute nature of German industrial bottlenecks meant there was a great deal of apprehension about the supply of these materials. This gave these neutrals some leverage so long as they supplied Germany with raw materials, they could keep their neutrality and replenish their stock of hard currency. Here was where these neutrals' military strength and geography played a role in that invading these areas would have cost the Third Reich men and endangered these sources of vital material.

Of course, Germany's need for natural resources was only one part of the picture. The overall course of the war affected Germany's attitude towards neutrals. For example, when the Anglo-French had started to pressure Sweden to stop iron ore exports to Germany in late 1939/early 1940, the German invasions of Denmark and Norway were proactive measures to prevent this from happening. Germany's attitude towards the remaining neutrals cannot be separated from larger wartime context, especially the invasion of the Soviet Union and the general collapse of Italian military fortunes. The size and scale of Barborossa put a great strain on the overstretched German military establishment. This meant that there was very little slack for expanding military operations and there was a persistent fear from late-1941 onwards that German troops would have to fill in the vacuum of their allies' shortcomings. In a memorandum on final victory, OKW contended that with regards to its allies and neutrals:

Germany would succeed, by political means, not only in inducing its allies to intensify their war efforts, but also in securing the periphery by bringing the flanking powers-hitherto neutral-of Turkey, Spain, Portugal, and Sweden into the continental defensive bloc. {emphasis mine}

In short, OKW's hope was that these neutrals would continue to supply strategic war material to Germany and resist entreaties by the Allies to join the war. This German overstretch put a damper on any enthusiasm for courting Franco to join the war since although Spain's army was relatively large and battle-hardened, its equipment was approaching obsolescence and Spanish military operations would dip into Germany's already precariously short stocks of fuel. When Italy finally did collapse in 1943, there was a fear that not only would the Allies capitalize upon the power vacuum, but that Turkey would join on the side of the Allies. this provided one of the impetuses for one of the last German victories in the war, the Dodecanese campaign in September-November 1943 in which Germany occupied Italian-held islands and defeated Allied reinforcements to the Italians in the Aegean. The expansion of the war in the Pacific also heightened Germany's wider diplomatic isolation and keeping these flanking powers neutral was highly important for German intelligence and diplomatic corps to have contact with the wider world, however circumscribed.

Finally, it is important to realize that Germany's long-term plan was to win the war. Hitler's strategic vision of a German-dominated continent fusing economic and racial hegemony was very expansive, but it also one that very few non-German powers could get behind. This lack of any workable level of European partnership encouraged German planners to think purely in terms of the short-term immediate gains. For example, there were some industrialists within the Low Countries and France that were amenable to the new economic order in 1940, but Germany preferred to plunder these territories' industries for the Reich, where much of the material languished due to the shortage of skilled labor. The shortcomings of German industry encouraged this type of thinking as the neutrals provided an immediate need and were not making moves to attack Germany. As it became apparent that Germany was going to lose the war around 1943, German policy towards the neutrals became more conciliatory and desperate to keep whatever clout they still possessed, which was swiftly evaporating.

Sources

Bowen, Wayne H. Spain During World War II. Columbia: University of Missouri Press, 2006.

Mazower, Mark. Hitler's Empire: How the Nazis Ruled Europe. New York: Penguin Press, 2008.

Militärgeschichtliches Forschungsamt. Germany and the Second World War Volume III. Oxford: Clarendon Press, 2001.

_.Germany and the Second World War Volume VI. Oxford: Clarendon Press, 2001.