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In Terry Tao's blog, specifically this post, he says that nilpotent groups can be thought of in algebraic geometry and analysis as modeling infinitesimal perturbations of the identity. Can someone provide a reference where this point of view is used?

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If you can read french, have a look at this. numdam.org/numdam-bin/browse?id=PMIHES_1965__26_ –  Baby Dragon Jul 3 '12 at 7:36

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The idea of nilpotents representing infinitesimal change dates back at least to the invention/discovery of calculus by Newton and Leibniz! In deformation theory of schemes at least, the whole story begins with the study of flat families over the spectrum of $k[t]/t^2,$ which are viewed as infinitesimal perturbations of the "central" fibre parametrized by the nilpotent parameter $t.$ This is mentioned in Hartshorne II.8 in the exercises I believe, but there is an entire field of work in deformation theory dedicated to this beautiful concept. For example, Harthorne has a recent introductory book called "Deformation Theory".

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