We are wanting to see Jungfraujoch, Titlus and the Matterhorn (if we get lucky) in a five day trip coming from Ulm in Germany and leaving to travel on to France. We are not renting a car. Which would be the best home base?
You are wanting to see places in very different areas. If you want to stay in Murren, enjoy the things around Murren. If you want to stay in Adelboden, enjoy the things around there. Trying to hit all the major tourist highlights in Switzerland from one base is a recipe for spending lots of time on trains and not as much time enjoying the place you're at. Murren will help you see Jungfraujoch, but you'd want to skip Titlis & Matterhorn and substitute them for other amazing things to do in the Jungfrau region. Adelboden is less on the tourist radar, which might appeal to you, but wouldn't be convenient to anything on your list. If you want to see the Matterhorn, stay in Zermatt. If you want to see Jungfraujoch, stay in Grindelwald, Wengen or Murren. If you want to see Titlis stay in Engelberg or Luzern. They are 3 distinct areas.
Also note that Adelboden is not on the train line, so you must take a bus from Frütigen to get there.