I love it! Do you put them on your shoulder and walk through stores or around the park too? Or do you usually have the mobile tree unit thing for them to hangout on and let them chill on that while you take care of things??
When the O.P. "psylvia" said "He goes on outings with my daughter, happily perched atop her head" -- I kind of envisioned your outings / errands the same way: with a chameleon perched on your head or on your shoulder. That's kind of what I was getting at I guess when I asked if it stresses them. In most of your pics though, your chameleons are chilling on that sweet little vine tree stand. Do you actually take them into stores and things of the such too, such as through the grocery store like the O.P. does? THAT'S what I would figure would stress them a bit with all the walking around on a shoulder instead of a branch, unfamiliar faces walking by you in the stores or at parks, unnatural environment in stores.....
That doesn't cause them to freak out? I know traveling freaks my little man out a bit, but he isn't used to it like yours is. And I've NEVER even TRIED to walk around a store or a park or anything with him. It never really crossed my mind. But yours can do all these things... and not freak out?
If that's the case, then maybe the common belief that chameleons don't like to be handled a lot doesn't hold as true as everyone is made to believe. I handle my chameleon minimally, only once or twice a week to go out for sun, and I can clean the cage while he is out too. When he was younger, I handled him a bit more often so that way he would get used to me and know I am not a threat. As he grew, I slowly began to handle him less and less so that way he can be nice and happy because I, just like many others, have read time and time again that chameleons should not be handled regularly because it stresses them.
If you go out and about with your chameleon a lot and they are used to it and don't care, then that opens a whole new chapter in the world of chameleons. It shows that they CAN like it, if they become accustomed to it. I, like many others, have always had the belief that they like to be left alone most of the time, but you sound like you spend LOTS of time with them and they love it! That's really neat. I guess they can become more tame than previously thought.....