I think it would have to play well with Taylor series -- for example e^s would be 1 + s + s^2/2 + s^3/6 + s^4/24 + ... -- and that would be mixing units. That's not to say it's meaningless, but rather you'd have to accept measurements like 4s + 5s^2, with incompatible units hanging around. (There's something similar in geometric algebra, I think, where expressions contain terms of different dimensions added together.)

Interesting point. But I think that the Taylor series is just a mathematical tool, and as such it hides the units that exist in these expressions. From a physics point of view, every term in the Taylor series needs to be augmented with a unit that follows from the definition of the Taylor series.

This is purely a guess, but I think it's because all the exponents are scalar. You get exponents when the derivative of a function is proportional to itself:

    d/dx (f) = k*f

So exponentials will be scalars because the exponential part is scaling the original value.

Example: population growth is exponential.

    d/dt(p) = (ln(2)/r)*p
    p(t) = p0 * 2^(t/r)
    p : population
    p0: population
    t : days
    r : days (time it takes for the population to double)
    t/r : scalar