Probably my favourite LA adventure game, though Sam & Max really does give it a run for its money. I played it at what I suspect was a very formative time of my pre-teens, and I think it did a lot to shape not only my humour, but also my understanding that games could be challenging yet satisfying in how they deliver a story through puzzles and obstacles. I think it struck a very good balance between a hand-holding push-over and being so challenging that you needed to dig out a cheat guide every few minutes. That's largely a credit to its story-telling and design, and I think I was just getting to the age where I was able to appreciate that.
I'm 35 years old and had a similar experience. What you said about it shaping your humour really struck a chord with me (particularly Sam & Max). Maybe because it was the first LA game I played, but I think DoTT is the best, but it led to me playing all the rest.
I got the talkie on CD-ROM for a christmas present one year. I struggled with a lot of it, but my Dad before giving me the present removed the hint/solve book from the box and watched us (me and my younger brother) play and try and figure everything out and just gave us enough of a hint at the right times to make it feel like a really satisfying experience.