I cannot find any information on Google or Wolfram Mathworld to answer this question. I also don't have the skills to calculate it myself so I thought it would be good if someone with this knowledge could share it here.
2026-03-26 21:25:57.1774560357
What is the simple continued fraction of $τ$ ($2π$)?
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$$2\pi \approx 6+\cfrac{1}{ 3+\cfrac{1}{ 1+\cfrac{1}{ 1+\cfrac{1}{ 7+\cfrac{1}{ 2+\cfrac{1}{ 146+\cfrac{1}{ 3+\cfrac{1}{ 1+\cfrac{1}{ 138+\cdots}}}}}}}}}$$
In How to find continued fraction of pi I explained how to calculate a simple continued fraction, using $\pi$ as an example. It's straightforward arithmetic and does not require any theory.
Once we have a few terms, we can search for them in the Online Encyclopedia of Integer Sequences, which produces sequence A058291, “Continued fraction for 2 Pi”. This page gives 97 terms, has a link to a listing of 20,000 terms, and other links to more information.