Hi, Bruce:
Rearding the very same sequence you mention, I quote from the nice discussion of Euclidstyle sequences in Crandall and Pomerance:
Quote:
The sequnce {insert your sequence here} was considered by Guy and Nowakowski and later by Shanks. [Wagstaff 1993] computed the sequence through the 43rd term. The computational problem inherent in continuing the sequence further is the enormous size of the numbers that must be factored. Already, the number q1...q43 + 1 has 180 digits.

So this "puzzle" is in fact an unsolved research problem  I suggest you combine a suitably edited version of my post with your PM to me and start a thread in the Math section.
Amazing how quickly one can get to deep, interesting research topics in number theory merely by playing around a bit with Euclid's original idea.
Cheers,
Ernst
BTW, if you don't yet have the book by C & P, I highly recommend it.