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54 17 Sequences that avoid arithmetic progressions have been studied, e.g., "Sequences Containing No 3-Term Arithmetic Progressions," Janusz Dybizbański, 2012, journal link. I started to explore sequences that avoid both arithmetic and geometric progressions, i.e., avoid $x, x+c, x+2c$ and avoid $y, c y, c^2 y$ anywhere in the sequence (not necessarily consecutively). Starting with $(1,2)$ , one cannot extend with $3$ because $(1,2,3)$ forms an arithemtical progression, and one cannot extend with $4$ because $(1,2,4)$ is a geometric progression. But $(1,2,5)$ is fine. Continuing in the same manner leads to the following "greedy" sequence: $$1, 2, 5, 6, 12, 13, 15, 16, 32, 33, 35, 39, 40, 42, 56, 81, 84, 85, 88,$$ $$90, 93, 94, 108, 109, 113, 115, 116, 159, 189, 207, 208, 222, ldots$$