I'm heading to Europe in late September for a four week trip. I'm a 24-year-old male looking to immerse myself in other cultures and explore new cities. I enjoy art and architecture, food and drink, and night life. I'm also content to spend some time sitting in a cafe doing some writing.

Here's the places I'm definitely going: London (~5 days), Amsterdam (~3 days), Berlin (~5 days), Prague (~3 days), Budapest (~4 days), Munich (~3 days). Adding in some time for travel, that leaves me with a few days to visit another destination.

Right now I'm thinking either
1. Belgium (seeing some combination of Bruges, Brussels, Antwerp or Gent)
2. Vienna
3. Hamburg

I know these are three wildly different places to compare, but based on the other places I'm seeing and your personal experiences, what would anyone recommend? Alternatively I could pick two of them and shorten my stay elsewhere. Or not see any of them and lengthen my stay elsewhere. All that is set in stone is that I'm flying into London and flying home from Munich.

Thanks in advance for your advice!

Consider Salzburg which is about 2 hours by train from Munich. Or better yet Lucerne although that may be too far out of your way. I would rather visit either in lieu of Vienna or Hamburg.

You will certainly have a wonderful time no matter which of these you choose. However, just a thought to consider.... Since everyplace else you are visiting is a large-ish city, consider building in one non-city destination. A few days exploring the country side, or hiking a mountain, or discovering a small town might provide a nice contrast to the more urban destinations. You could bike in Beligium, hike in Germany or Austria, stay on a farm, etc. Think about whether you want a solid month of cities, or would like to see a little of one of the many other sides Europe has to offer.

I agree: try to get out of the big cities for part of your trip.

You might throw in something like Slovenia or Croatia to get off of the "American tourist trail." easyJet may have cheap flights to either from London if you book in advance.

Get out of cities!
See the Salzkamergut, Hallstatt, and Cesky Krumlov.

If you are seriously interested in architecture, I would add Innsbruck. As unlikely at it may seem, they have been rebuilding their structures and doing outstanding work, from the ski jump by Zaha Hadid to the David Chipperfield modern shopping mall between 18th century palaces. Even small homes in the suburbs and M-Preis supermarkets in the country are often outstaning architecture works.
Also Barcelona may be interesting on the architecture side, but it is too distant from your other places.

Stay in the cities!
Lie on a beach!
Stay on a farm!

Wait....do what you want. And it looks like you have actually been specific about your interests and options. SO I will give you your first response that is focused on what you asked for instead of what I would like to see you do.

Given the locations on which you asked for opinions, Hamburg excels for night life, and lags on architecture and art. A fascinating city but not a classically beautiful city.

Vienna probably wins on art and architecture, loses on the night life. The cafe scene was pretty much born there, but you may not be thinking of traditional indoor Viennese cafes when you think of a place to sit and write. Gorgeous and massive in it's scale of architecture.

Belgium could give you some of everything depending on how you split up the cities and would introduce some different food opportunities into the mix. Brugge is an atmospheric gem. Early morning and late night walks around the canals can be very special.

Consider Rotterdam for modern architecture, it will make a change at any rate. You could of course add time to some of your cities. Rotterdam could be visited as a day trip from Amsterdam, if you were to add a day or two to your stay there.
You could also visit some other towns/cities/countryside/beach as day trips too by adding days to your chosen cities.

Hamburg very good, Hanover good. Munich only borderline in my mind

I wouldn't add another destination. I vote "not see any of them" and add that little "extra" time divided up among the places you are already planning to visit. Amsterdam could easily use more than only 3 days.