Actually if you think in the sense of informatics...base 3 is more efficient than base 2, but natural base (base e = base 2.718...) is the most efficient, and 3 - e is closer to e than e - 2. So it is more natural for computers to go in multiple bases but unnatural to human.
There is a particular notion of "radix economy" which takes into account both the length of a number compared to its size and the informational complexity of more symbols. It doesn't really have anything to do with data storage though. If you could create a device to store 1000 10-bits (10its?) for exactly the same cost as one that stores 1000 3-bits you'd obviously pick the base 10 one, no matter what radix economy says.
What does it mean to have a transcendental base? I understand algebraic bases but I don't see how you could use a transcendental base to represent integers with a finite number of symbols
The idea is that if it costs $r to store a base-r digit, then base 3 (or e in a continuous scale) turns out to be the most efficient. Obviously, there's no a priori reason to think that a 3-level gate is exactly 1.5x more expensive than a 2-level gate, so this is mostly of theoretical interest.
I'm thinking about how this would apply to human psychology of reading and writing numbers. Then it doesn't make sense to measure economy as b floor(log_b(n)+1), because adding in more symbols doesn't increase the complexity linearly for people reading or writing numbers. Maybe something like E(b,n) = f(b) g(floor(log_b(n)+1)), where f stays constant up to 10 or 20 symbols, and then increases after, and g increases faster than linearly because it's easier to read shorter numbers than longer ones.
