Thanks for all your suggestions, BabyGrand. The idea of playing in public (in fact, in front of anyone) would be a huge problem for him. Even playing for one person would be very stressful, which is why he doesn't want to take any practical exams. I'm pretty sure that it would also deter him from getting together with my other adult pupils. As he got to know me better, he eventually stopped shaking in his lessons and realized that I am a very calm and relaxed teacher and that I never put any pressure on him. Although he still berates himself when he plays a wrong note.
I often ask him if there's anything special he wants to play and have introduced him to various kinds of music. He very much likes playing Martha Mier pieces but I think he feels that he shouldn't only play pieces that he enjoys! It's quite a complicated situation and I'm giving a lot of thought as to how best to help him. Again, thanks to everyone who has offered suggestions - I really do appreciate it.
I have some adult students who do not want to perform in front of others or take an exam etc, who still love coming along to our group events. There is never any pressure to perform in front of the group. It's like a cross between a group class, an ensemble and a social gathering; it's definitely not a concert! We do a lot of playing together as a group (this is with a mixture of piano and violin students, and sometimes singing students too), along with learning and discussing things together. I let them suggest topics, and we have covered things like improvisation, how to play with others, practice strategies, performance nerves, playing in different styles, etc. When there are 5 or 10 people all playing at the same time, no-one feels "exposed"....it's a very safe environment. Duet lessons are also popular with my performance-shy students - they feel ok because they are playing at the same time, not in front of one another.
I have also been surprised in the past - when I've asked the group if anyone would like to play something for the others, and a student who had incredibly severe anxiety issues and had told me she never wanted to play in front of anyone piped up "Ok, I will"! She still would never want to do exams or play in a concert, but she played for our little group. I would never have dreamed of asking her to do it, but I was delighted that she felt comfortable enough to want to.
I also agree with BadStrad. Maybe your student needs to be given "permission" to just enjoy his playing - to enjoy where he's at for a while, without feeling any (self-imposed) pressure to progress. It's fine for him to play some music he likes for a while, and it's fine for him just to sit at his piano and play pieces he loves because he loves playing. Maybe it would help for him to know that you actually would love to hear that he had gone to his piano and just sat and played for pure enjoyment, rather than him feeling that he's somehow letting you down / not doing what he's "supposed" to / being lazy etc. He doesn't need to feel bad or guilty about where he's at. If the best thing for him right now is to play the music he loves, then that's ok. There's no rule that says he has to be doing X, Y or Z. Of course you know that, but maybe he needs to hear you say it.
On a practical level, if he loves Martha Mier but feels he "should" be trying other composers, could you try some music in a similar style by a different composer? There's quite a wealth of material out there. Maybe just having a new book would feel like a new challenge? I have a lot of this kind of music on my shelves....roughly what sort of level is his playing? And I can see if I can suggest some options.
I love teaching my adult students, but they are definitely a lot more complex than children! It's a great challenge for us as teachers to work out how we can best help each of them.