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I know it's not possible to perfectly, trisect an angle with compass and straightedge, but what's the closest anyone has gotten to doing this?

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    You'll need to be more specific. It actually is possible for some angles. One can artificially trisect 90 degrees, e.g. – 2017-01-17

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It's easy to "quadrisect" an arbitrary angle by bisecting it twice. You can get arbitrarily close to trisecting the original angle by repeatedly quadrisecting the angle and adding together the fractions that you generate, since $$ \frac13=\frac14 + \frac14\cdot\frac14 +\frac14\cdot\frac14\cdot\frac14+\cdots. $$

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    That's very complicated. The methods George described below are much simpler. – 2017-01-18
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    ...I think I want to know your definition of "simple." – 2017-01-18
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    @G. G. G. The point is that this method can come _Arbitrarily close_ to trisecting the angle with a finite number of steps. The questioned asked for the "closest method". Given enough iterations of this process, _any_ other method can be eventually surpassed. – 2017-01-18
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I dug up the following method from the 1966 Los Angeles Times:

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Both methods work, but I'd say the one at the top is best.

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    Amusing as it may be, this is not an answer unless/until you write down a proof. (If you believe that newspaper ad enough to waste time on it, that is.) – 2017-01-18