I would love to play a campaign with no combat at all but throughout my decades of playing, at least 90% of the other players ache for regular battles. Few could sit through a basic Call of Cthulhu scenario and actually do simple detective work, interview people and find crucial clues without arming up and wanting to fight something. Even the GM realizes that we have to toss them a combat bone on a regular basis to keep them interested.
I've never run an entire campaign without hack n' slash, but I've had several sessions that were non-combat. There is certainly plenty of fun to still be had. For one group, I set up a scenario for two level 1 PCs. One was lowest rank in the thieves guild (which may or may not exist) and one was hoping to get into the guild (that absolutely, totally, does not exist).
The initiate is contacted via a note from a higher ranking member because he's been observed picking pockets or petty thieving and is told that he may be a good fit. There's going to be a challenge for initiates and the winner gets taken aboard as a member. The initiate has no idea how many are competing nor who is also in the challenge. All initiates will have to take a specific item from a tavern at one end of town and bring it to a tavern at the other end of town without being caught and there can be no killing. One mustn't draw attention to the guild. The initiate to succeed is let in.
The existing probationary-level guild member is assigned an initiate and his job is to aid his initiate in succeeding. If the initiate finds out he is being helped, both are disqualified. If either is caught, both are disqualified. If the member's initiate succeeds, the member is moved from probationary member to lowest level member.
What the PCs don't realize is the the actual challenge is for the existing low-level member. The initiate is just a fall guy should something go wrong. If the initiate is caught, he has no ties at all to the Guild and nothing can be betrayed.
We had a 'falling out of our seats, tears in our eyes laughing session' that lasted a good 2 hours. At level 1, the necessary thieving skills are all low, if they even have the skills they need. I wanted the Thieve and Rogue PCs to get an idea of the skills they should be investing in if they wanted to play those professions. I ran through all of the non-combat skills RMC-II had. Stalk/Hide, Sense Ambush Assassin, Hide Item, Duping, Acting, Lie Perception, Sprinting, Disguise, Acrobatics, Tightrope Walking, Grappling Hook, Rope Mastery, Pick Pockets, Surveillance, Improvised Weapon, etc. Obviously they couldn't have all of those skills, so there were plenty of -25's and failed attempts and much hilarity ensued.
The initiate wanted to distract an NPC by throwing a mug. He fumbled the throwing skill spectacularly and creamed the tavern owner's wife in the face and knocked her out. He turned to run, fumbled his Sprinting skill and fell flat on his face. A couple of thugs were on him and he rolled very well for Acting and was able to convince the thugs it was another patron "It was him! Look! He's the only one here without a mug of ale! It was obviously him!!!"
Then the fight started...