Any real number x has an infinite continued fraction representation. By efficient I mean that the information of the continued fraction coefficients is an efficient way to compute rational upper and lower bounds that approximate x well (they are the best rational approximations to x).
> They are occasionally convenient for certain problem structures but, as I'm sure you've already discovered, somewhat less convenient for a wide range of common problems.
I'm curious what you mean exactly. I've found them to be very convenient for evaluating arithmetic expressions (involving both rational and irrational numbers) to fairly high accuracy. They are not the most efficient solution for this, but their simplicity and not having to do error analysis is far better than any other purely numerical system.
> fundamentally equivalent to simple fractions.
This feels like it is a bit too reductionist. I can come up with a lot of example, but it's quite hard to find the best rational approximations of a number with just fractions, while it's trivial with continued fractions. Likewise, a number like the golden ratio, e, or any algebraic number has a simple description in terms of continued fractions, while this is certainly not the case for normal fractions.
That continued fractions can be easily converted to normal fractions and vice versa, is a strength of continued fractions, not a weakness.