GERMANY

I've gone twice. Both times in late Sept. early Oct. The weather is great and the leaves are just starting to change. If you like beer and fun Oktoberfest is going on and I would highly recommend it if that interest you. It's a faire on steroids with specially brewed Oktoberfest beer that is very tasty. Even if it's not your thing, the town is very lively during that time period.

A few years back I was in Germany from September - December. Sept and Oct were really nice with cooler weather, leaves turning on the hillsides, and spotty rain. November was quite cold, leaves were gone from the trees, and it rained quite a bit. December was quite magical with dustings of snow and Christmas markets in every town. The Germans know how to do Christmas!

Define "best" for your priorities, and perhaps we can provide more detailed answers.

When is the best time to visit the United States? The answer is the same as for visiting Germany. It depends on what you want to do, cities, small towns, winter or summer sports, sight-seeing and so forth. The weather is always a chance. It can be hot in March and cold in July.
A more specific request will get you better answers.

I've spent 4-5 month (cumulatively, 12 trips) in Germany, all but three months (Feb, Jun, Jul). Many of my trips have been over Labor Day (Aug-Sep) or in October. I've always run into good weather those months. I'm going back this year in Sep-Oct. According to a chart in ETBD, October is the driest month in Bavaria.

Joan and Lee are both correct. Weather in Germany can vary greatly from year to year, but September is usually the most stable. Usually relatively dry, warm but not hot.