Quote:...in theory, could it exist?
That really depends on what your definition of "exist" is. (Sorry Bill Clinton.)
The problem with using 1 as a base is that it wouldn't match the rules for any other base. For example, we know that the third place left of the decimal in base 2 is 2<sup>2</sup> or 4. So, 101<sub>2</sub> = 5<sub>10</sub> . But 1<sup>x</sup> = 1, which means every place has the same value.
Also, 0 would be the only valid digit. So the only number you can really express is zero.
Now, in Set Theory (which I only took because I was a Math major, but I loved it), the numbers we know and love (the Natural Numbers), are defined in terms of something very similar to this. Zero is the empty set. One is a set containing zero. Two is a set containing one and zero. And so on. Each number is a set with the right number of elements, and all the elements are defined recursively from NOTHING. This is where most people say "stop, I wanna get off."
Anyway, you could theoretically change your definition of base to include stings of zeroes, and the number of zeroes would determine the size of the number. But it's not really the same as other bases.
Where I tend to jump off the train is when people start talking about base e or base pi. No thanks -- I like my bases to be positive integers (greater than 1).
"Bad news, bad news came to me where I sleep / Turn turn turn again" - Bob Dylan