TITLE: What's the closest approximation to $\pi$ using the digits $0-9$ only once?
QUESTION [32 upvotes]: What's the closest approximation to $\pi$ achievable using each digit $0-9$ no more than once, and basic operations of roots, brackets, exponentiation, addition subtraction, concatenation, division and factorial?

This was mentioned in another question and I thought it was fun.  I can pretty quickly come up with (with a bit of help from Ramanujan):
$$\frac{7}{3}\left(1+\sqrt{\frac{6}{50}}\right)-\frac{8}{2\times4\times9!}\approx3.1415958$$
I challenge anybody to get beyond 20 decimal digits of accuracy!

REPLY [5 votes]: A simple exact expression using the gamma function: $$\left(\frac63\cdot\left(\frac48\right)!\right)^2$$
My answer below, based on tehtmi's answer, has two more decimal digits of accuracy. It applies a factor to the sum of the first five terms instead of adding two more terms.
$$(\sqrt{^5 70} - \sqrt{^{15} 9!} + \sqrt{^{23} 8!} + \sqrt{^{34} 14!} + \sqrt{^{36} .5}) * \sqrt{^{44} 6} * \sqrt{^{50} 2} * \sqrt{^{58} 3} \approx \pi - 1.926.. \times10^{-18} $$
The equivalent expression here can be plugged into Wolfram Alpha: (70^(.5^5)-9!^(.5^15)+8!^(.5^23)+14!^(.5^34)+.5^(.5^36))*6^(.5^44)*2^(.5^50)*3^(.5^58)