Exponential and Continued Fractions

The continued fraction expansion of a real number is a fundamental and revealing expansion through its connection with Euclidean algorithm and with ``best'' rational approximations (see [HW]). At the same time, it is very poorly understood for some interesting numbers. We know that it is essentially unique and finite (i.e., terminating) exactly for rational numbers and periodic exactly for quadratic irrationalities. But apart from that, the expansion of even a single additional algebraic number is not explicitly known; we do not know even whether the partial quotients are unbounded for such numbers. (See [BS] for the function field situation). For transcendental numbers of interest, it is not clear when to expect a continued fraction with a good ``pattern''. For example, Euler gave a nice continued fraction for e