Dear Alex,

It seems to me that the general question in the background of your query on algebra really is the better one to focus on, in that we can forget about irrelevant details. That is, as you've mentioned, one could be asking the question about motivation and decision in any kind of mathematics, or maybe even  life in general. In that form, I can't see much useful to write other than  the usual cliches: there are safer investments and riskier ones; most people stick to the former generically with occasional dabbling in the latter, and so on. This, I think, is true regardless of your status. Of course, going back to the corny financial analogy that Peter has kindly referred to, just *how* risky an investment is depends on how much money you have in the bank.  We each just make decisions in as informed a manner as we can.

Having said this, I do rather like the following example: <a href="http://en.wikipedia.org/wiki/Generalized_Cartan_matrix">Kac-Moody algebras</a> could be considered 'idle' generalizations of finite-dimensional simple Lie algebras. One considers the construction of simple Lie algebras by generators and relations starting from a Cartan matrix. When a positive definiteness condition is dropped, one arrives at Kac-Moody algebras. I'm far from an expert on these things, but I have the impression that the initial definition by Kac and Moody in 1968 really was somewhat just for sake of it. Perhaps indeed, the main (implicit) justification was that the usual Lie algebras were such successful creatures. Other contributors here can describe with far more fluency just how dramatically the situation changed afterwards, accelerating especially in the 80's, as a consequence of the interaction with physics, that is, conformal field theory and string theory. But many of the real experts here seem to be rather young and perhaps regard vertex operator algebras and the like as being just so much bread and butter. However, when I started graduate school in the 1980's, this story of Kac-Moody algebras was still something of a marvel.
There must be at least a few other  cases involving a rise of comparable magnitude. 

Meanwhile, I do hope some expert will comment on this. I fear somewhat that my knowledge of this story is a bit of the fairy-tale version.