I have been using Mathematica for about ten years. I have found the design philosophy of Mathematica to be useful for not just performing difficult analytical derivations, but also for manipulating data of all sorts. Where MATLAB is based around the matrix, Mathematica is based around the function. Algebra, Mathematica's forte, is a process of pattern recognition followed by manipulation of functions and their arguments. Geometry, text, and numeric files all can be abstracted as nested functions, and therefore can be manipulated "algebraically." The power of this cannot be overstated. Mathematica can analyze information of all formats from abstract theory to measured results. Therefore it is usually the first and last tool I use in every project where the analytical derivations, the simulation proof-of-concept, and the experimental results get integrated to yield the final figures.
I consider myself an expert in Mathematica and I have hundreds of files that I've authored for various purposes. Some of the more interesting things I've done with it are: proximity correction for e-beam lithography, solving non-linear differential equations, overlaying 3D geometries with 2D graphics, finding optimal LED blinking patterns that appear smooth, simulating attack strategies in a "dog-fight", translating code from one computer language to another, finding analytical solutions to generic n-layer electromagnetic boundary problems in cartesian, cylindrical, and spherical coordinate systems, finding far-field diffraction patterns from apertures, parameter extraction of complicated optical transmission and reflection data, processing ultrasound audio files, learning how detector nonlinearities effect FTIR measurements, finding the induced dipole on a gold nanorod, post-processing AFM images to remove artifacts, processing diffraction patterns to determine aperture properties, and solving for the deflections in timber-frame structures with various joint structures.