This has already been studied in the context of Moore's law [0]. The answer is that if you assume a doubling every 18 months, then any project that takes less than (18 / ln 2) ~= 26 months will finish fastest if begun immediately. For a project that takes longer, you should wait until your project will take 26 months to complete with current hardware, and buy your new hardware and start it then.

You can plug in whatever numbers you want for the growth rate, but I don't think there are many uses of LLMs that require anything like months' worth of tokens to generate results at this point. Feel free to slack off if you are one of the exceptions.

[0] https://arxiv.org/abs/astro-ph/9912202