Question is pretty much in the title. What is the greatest area that Germany (including das Deutsche Reich) ever covered. I'd expect it to be either during the World Wars or during the colonial era, but I couldn't find the numbers. Yet I might be wrong and there might be another answer.
Depends on what you mean by "Germany". In terms of modern ethno-national nation-states, the "core territories" of the Third Reich were the largest any modern German-language nation-state ever was, with the total area of the 1939 borders plus Alsace-Lorraine and the Danzig corridor being 633,786 square kilometres. The German Empire was a close second with its European territories equalling 540,857.54 km². However, if you include the German Empire's overseas colonial holdings, the German Empire would have an area of about 3.2 million km².
Taking the concept of "Germany" more loosely and applying it to include all German language polities, Austria-Hungary had larger European holdings than the contemporaneous German Empire, with a total area of 681,727 km². I've been unable to find an exact estimate of the total area of the German dominated Holy Roman Empire at it's territorial zenith in the 12th century, but it was larger than either the German or Austro-Hungarian empires in terms of European holdings, albeit with large swaths of their territory speaking Romance and/or Slavic languages.
Apart from what you mean by "Germany", the answer will also depend on what you mean by "controlled". Assuming that your answer to the first question is present-day Germany and its immediate nation-state predecessors (ie. Hitler's Third Reich, the Weimar republic and the "Deutsches Kaiserreich" founded in 1871, but not the Holy Roman Empire that is at least problematic in terms of "German-ness"), and assuming that "control" in your question means "military control", the answer to your question will probably be the territory occupied by German forces and its allies in late 1942 (i. e. before the Soviet advances that led to the encirclement of the German 6th army.) At this point of time, German military or its allies were present in the whole of Europe except the Iberian peninsula, the British isles, and Sweden; in most of the (Vichy-) French northern African territories and within the Soviet Union eastward to a frontline running roughly from St. Petersburg in the north via just west of Moscow to Stalingrad and the Volga and Ural rivers in the south.
By visual inspection of the globe, this should be somewhat larger than German pre-WW I colonial holdings in Africa (southeast, today Tanzania / Mozambique, southwest, today roughly Namibia, and roughly what is Cameroon today) and Southeast-Asia (mostly smaller isles and some territory in South China). According to this German wikipedia entry German state territory and colonial holdings totalled an area of 2.7 million km², while the above sketched area occupied during WW II should be something of 8 - 10 million km². (By comparison with the total area of Europe to the Ural mountains of 10 million km², precise numbers are harder to obtain, as the eastern European frontlines don't reflect today's state borders, so you can't just look up the areas of the states in question.)