WordPress is still a good piece of software, so I'm in no way recommending against using it. Just be aware that many of the functions output markup and that you'll have to filter them out. WP has some functions that allow you to do that, though it's not always a simple process, I find.
My favorite CMS is ExpressionEngine. It is commercial, however, but you have 99.9% control of all your markup and don't need to touch a single line of PHP unless you'd like to get really advanced and do something that either isn't built into the system already or isn't provided by a third-party add-on. It's not inexpensive but you get a lot for your money.
Two other really great ones that give you more than 99% control are Perch (commercial) and Statamic (commercial).
There are two left that I recommend. The first being MODx Revolution. There is a learning curve, however, but well worth investing into. It's open-source, so doesn't cost you anything except time and effort.
And finally, if you want a really simple open-source CMS, I can recommend Textpattern. It's a very sophisticated little CMS. It's clean, lightweight, logical, simple, conscise, and user-friendly.
Frankly, it is not as popular as any of the others I mentioned above, but that has nothing to do with its quality. Textpattern is a quiet contender and there are quite a few professionals using it for their own sites. I used it when I started out and only moved away from it because I found ExpressionEngine and liked that even more.
For basic stuff, like a blog or magazine, Textpattern is perfectly apt. For more complex sites, you might want to try something else, however.
I should also mention Drupal and Silverstripe as they seem very popular and get recommend a lot as well, though I can't say anything about them as it's been years since I last worked with Drupal and I've never even tried Silverstripe.
And here are demos you can try out for each of the above:
Statamic: Doesn't have a demo but a few videos that show you what it's like working with it: http://statamic.com/learn/screencasts