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OK, I may be wrong here because there could be some research going on that I haven't read about yet, but the simple answer is no.
Binary is commonly thought of as 1s and 0s, but it is in reality (in most cases) voltage on and voltage off, hence why computers are known as the two state machine. Of course it isn't only voltage, it can apply to the tiny pits on a CD or the holes in a punched card (going back a bit there).
In theory you could have trinary, with (for example) a cable where:
0V = 0
1V = 1
2V = 2
but there are a number of reasons why this is not used.
1. Binary degrades well (any voltage on a wire, whether the correct value or not, is known to be 1 or ON, similarly with a CD)
2. Two states are very easy to represent. CDs couldn't use trinary, as you can only have pit or no pit, not 'slightly less depth'
3. Binary is powerful as there are only 4 adding rules. Compared to 100(?) for decimal. Each time you add another representation (as you would be in trinary) you increase the number of mathematical rules and hence chips would need to cope with more and would be (in practice) less powerful.
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