[ugpbazfp] Practical "test" of irrationality

Suppose we want a random compactly expressible irrational number for aesthetic purposes: it is a portal to an endless stream of digits, peering into infinity.

To tell if a number is rational or irrational, calculate some continued fraction terms.  If the continued fraction does not terminate after some user-chosen number of terms, then declare it irrational "enough" for aesthetic purposes.  If it terminates, or has some very large terms, then it is rational or too close to rational for aesthetic purposes.  How many terms is enough terms for making the aesthetic decision?  How can we correctly calculate enough terms?

Consider randomly sampling a number from some collection of ways that typically generate irrational numbers.  We wish to avoid accidentally making a rational number.  Is the above proposed test for irrationality good?  That is, using ways that typically result in irrational numbers, how easy is it to accidentally create a rational number that has a very long (but terminating, of course) continued fraction representation?