Practitioners in signal and image processing heavily use the so-called Discrete Fourier Transform, which is a polynomial evaluated on complex variables. Applications in medical imaging abound.
The degree of these polynomials typically reaches the image size, which can be like 4096 or much more.
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Other consumers of polynomials are the error correction methods (such as those used in digital communication or CD readers). They are implemented in some types of barcodes, with degrees that reach dozens. If I am right, the QR codes use up to degree 69.