$\pi=3+\frac{1}{7+\frac{1}{15+\frac{1}{292+\frac{1}{1+\cdots}}}}$
In most years, π continued fraction day will be 3:7, giving the approximation 22/7, slightly better than 3.14. This year, however, it’s 3:7,15, the continued fraction giving $$\pi\approx 355/113$$, with an error of about $$2.7\times 10^{-7}$$. That’s two orders of magnitude smaller than the error in 3.1415, and it’s even smaller than the error when the US celebrates Correctly-Rounded Pi Day next year, 3/14/16