tldr; You're talking about 'part to whole relationships', here's most of the techniques for that currently in use.
Choose a visualisation method based on what message you're trying to bring out of the data you're trying to show, then style it as much as you want (being very careful to not get in the way of understanding the actual data and message).
You can use any image and chop it into appropriately-sized chunks, but be careful with that - it's a really common mistake to choose a cool-looking image or style then try to force data that doesn't fit into it. The end result seems positive at first, but doesn't actually work: people respond "Cool, that looks interesting! What does it mean? ... Erm, seriously, what does it mean?! Erm... Meh". Many people will be reluctant to even admit that they just don't 'get it'. Choose something robust that works, then make it look cool (without breaking it).
What you're talking about is referred to in the data visualisation world as a "part-to-whole" relationship. Here's a blog article on the topic from visual.ly.
The examples it lists are:
...which can be scaled to give an indication of total amount.
Pie charts (and actually, most types of chart that are good for part-to-whole relationships - particularly, anything based on area not length) get some stick from purists because it's difficult to see small differences or make purely numerical comparisons (e.g like "X is almost exactly double Y") - but if your focus and reason for visualising is to communicate the part-to-whole relationship, and you include the actual figures, that's fine. Nothing is better for accurate numerical comparison than actual numbers.
A rectangular area divided into chunks (and sometimes, sub-chunks).
Their example is a bit ugly, it's possible to make tree maps look much better than this, but it gets the idea across well:
Be careful with tree maps, they're not easy to do well. Having a logical order (largest to smallest) helps a lot.
For showing 'flow' with a part-to-whole relationship (their example sucks, here's a more illustrative simple one):
...and a more complex one:
Stacked bar, stacked area and 'parrallel sets' charts
For comparisons between equivalent wholes/sets, or different ways of slicing a whole or organising a set. 'Parrallel sets' charts are basically stacked bars that use connections instead of consistent order to show equivalence when moving between sets.
As for the general, "what else" question - of course you can use any shape if you're capable of accurately dividing it into chunks of the right sizes - but be very very wary of sacrificing the clarity of the information you are trying to show in an attempt to decorate it.
If the information isn't interesting enough in context on it's own, that's probably a problem with the information or the context you've set up, not the data presentation method. As a general rule, make sure your meaning is clear first, then make it pretty. Pie charts are popular because people just 'get' them - when used well of course; that intuitiveness comes at the cost that it's hard to see small differences accurately and that they become a horrible mess with more than about 7 segments (which is where a treemap can be better).
'Doughnut' charts are a popular way to make pie charts more interesting. They're just as clear (arguably clearer since there's no ambiguity about whether to compare by size of chunk or by angle), and it gives you space to play around, adding text, labels, icons, etc.
Never forget that conventions are really, really valuable things - the faster a reader gets how a visualisation works, the more time they spend focused on the meaning of your message, and the less time they spend scratching their head trying to decode the medium of your message.
More signal, less noise.
Oh, one other one for free :-)
Hierarchical, like treemaps, but circular and in reverse, so the 'leaves' of a hierarchical structure get more space and attention than the 'trunk'). Here's an example showing coffee flavours:
Just to show that it's not circles that are dull or over-familiar, it's what you do with them that counts :-)