Prague + what other cities?

As long as you don't have your return tickets yet, you can think out of the box. Look at GoogleFlights. Most are under $200 nonstop but it depends heavily on the exact date you are traveling.

Warsaw
Kiev
Odessa
Minsk
Belgrade
Bucharest
Ljubljana
Split
Istanbul
Riga

You don't say where you will be when you wind up your business commitments.

Berlin. lots of history and architecture (some very modern, some very old).

Easy to do from Prague is Dresden and Leipzig. Both are huge in terms of history. Also Berlin is only a few hours away.

It all really depends on what you're interested in.

I've always liked Prague as part of a Berlin, Dresden, Prague, Vienna, Budapest route. I think that's worthy of at least two weeks. Since Prague is in the middle, you could do one side or the other in one week - both sides have a city (Berlin and Vienna) with loads of sights that you may not be able to fully take in during your one week trip.

Why not rent a car and just drive around Czech Republic. Pilsen, Brno, kutna hora, cesky krumlov are all worth a visit.

I would suggest staying just in CR if Vienna is out this time. In CR see Brno (see Slavkov for the museum on Austerlitz and the Zentrum), Pilsen, and other historical towns.

If you are going to Prague for three days and have another four days to plan. I would try to stay in the general area. Most of the choices have already been mentioned.

I would save Vienna for when you do Budapest,.

You could see more of the Czech Republic or go to Krakow in Poland. Another choice is Dresden and Berlin. Another is Nuremberg, and some of the Romanic Road, including Rothenberg on the Tauber.

Travel less and see more. Dresden is just 2 hours away by train. The city offers several architecturally interesting buildings like the Zwinger, the Semper opera or the Frauenkirche. In the surrounding are wine villages, one of Europes biggest fortresses, a Renaissance "pleasure house" (= Lusthaus) with beautiful painting of birds, a huge Baroque hunting lodge and a palace with Chinoiserie elements which you can reach by a paddle steamer, just like Meissen with the famous porcelain manufacturer and a beautiful old ton topped by a castle and a Gothic cathedral. On a day trip you could see Görlitz, Germanys best preserved town, which is a must if you are into architecture and history, since it is also the only part of Silesia that wasn't awarded to Poland in 1945 (there is a Silesian museum in Germanys oldest Rennaissance burgher house). And if you like to hike visit Saxon/Bohemian Switzerland.