What is the reason for considering any algebraic structure? Because it comes up naturally when trying to do other things!
Here's a concrete example. In the Langlands programme one of the main local conjectures is relating representations of a (connected reductive) $p$-adic group to representations of a (group related to a) Galois group. Now most of the interesting representations of the $p$-adic group are infinite-dimensional, so this precludes one of the most powerful things that a representation theorist has in his arsenal---namely the possibility of taking traces. But in fact this can be fixed up very nicely! There is an analogue of the "group ring" of our $p$-adic group, namely the space of locally-constant complex-valued functions on the group with compact support. This space interits an addition (obvious) and a multiplication (convolution: the group has a natural measure on it, namely the Haar measure). So it's an algebra. Furthermore it is easily checked to have no identity element (the "delta function" isn't a locally-constant function!). However it's also not hard to check that there's an equivalence of categories between (certain) representations of the $p$-adic group that one is interested in, and (certain) representations of this algebra---the so-called Hecke algebra. Furthermore elements of the Hecke algebra act via maps with finite image, and so have traces! This is a big win. One can prove linear independence of characters etc etc, and get the powerful techniques back. But no way can the identity map be in this Hecke algebra---it certainly doesn't have finite image in general, and hence no trace.
Representations of the Hecke algebra are absolutely crucial in many works on this part of the Langlands correspondence, but they have no identity element. So there is one reason, in my area, at least.