Most of the time, I want games to challenge me. That can involve many different things. A lot of my favourites have heavy action and/or themes, and can have be tensed up and on the edge of my seat. I also enjoy more mentally challenging games, including puzzle-solving, or mastering complicated game mechanics or item management.
I find these things enjoyable, but I wouldn't call them relaxing. I wonder sometimes if a game can be truly relaxing. I have a very broad definition of what I will call a game, but I think some interactivity is required. And as soon as there is interactivity there will be consequences for actions, even if that's on an extremely basic level.
If I want a really soothing experience I'm probably not usually going to look to gaming. But, games can still calm me somewhat. I'm sure this is a very individual thing -- one person's relaxing is another person's slow and boring, after all.
My traditional "relaxation" game is actually Carmageddon 2. This is something like reading a trashy novel. I don't have to take anything very seriously, and just go with it. I just hoon around being a menace to pedestrians, and burn off some tension in the process.
In general, I think games with combat can be surprisingly calming, provided they are sufficiently forgiving or familiar. There is a point where you can get into the "zone" while fighting. There can be a fine line between that and grinding. These days I am wary of finding comfort in simple repetition, but I'm still guilty of it occasionally.
One game I've used for this purpose is PoPoLoCrois (PSP). This was an odd impulse buy for me, given my usual negative opinion of JRPGs. But I do have a weakness for fairytales, and was looking for something simple. It has very easy combat, so I didn't have to worry much about it. And (to me at least) it manages to be very cute without becoming cloying. The story was rather choppy, having been patched together from parts of older games. But overall it was a surprisingly charming experience. And charm can go a long way towards making a game experience relaxing, in my opinion.
Interestingly, I haven't found broad-appeal games without combat very calming. I haven't played The Sims or Sim City in many years, but I found the measured passing of time, combined with a lack of defined end points, had me very slightly on edge. I was always pushing, and trying to achieve something more.
Relatively recently, I have played more games that may actually have been designed with relaxation in mind. FlOw and Flower at least start out that way, with very floaty movement and atmosphere. But they'll sneak up on you, I find. In flOw the creatures become tougher and start fighting back, and even flower developed a darker feeling than I was expecting in places. They do remain suitable relaxation games.
This week I did acquire Endless Ocean 2: Adventures of the Deep (Wii). I gather they made an effort to make it a bit more focused and exciting than the previous game. But there is still a lot of simply swimming around looking at fish... at least if you are like me and interested enough to try and catalogue them. I'm sure to find enough entertainment there to justify the purchase.
I have probably not done my gaming credibility any favours today, but if there is one thing I look for in gaming it's diversity.