What is the smallest constant that has explicitly appeared in a published paper?

Here's a case where $10^{-2576}$ came up.

In 1928, A. S. Besicovitch introduced and studied regular and irregular $1$-sets in the plane (a fundamental notion in fractal geometry). He proved that the lower $1$-density of a regular $1$-set $E$ in the plane is equal to $1$ (the maximum possible value) at almost all (Lebesgue measure) points in $E$. To show how differently irregular $1$-sets in the plane behaved, Besicovitch proved that the lower $1$-density of an irregular $1$-set $F$ in the plane is bounded below $1$ at almost all (Lebesgue measure) points in $F$. The bound that Besicovitch obtained in 1928 was $1 - 10^{-2576}$.

In 1934, Besicovitch managed to improve this to $\frac{3}{4}$. [See Section 3.3 of Kenneth J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985.] "Besicovitch's $\frac{1}{2}$-problem" (presently unsolved, I believe) is to find the best almost everywhere upper bound for irregular sets. Besicovitch himself showed by a specific example that this bound is at least $\frac{1}{2},$ and he conjectured that it is equal to $\frac{1}{2}.$ In 1992, David Preiss and Jaroslav Tiser proved the bound is at most $\frac{2 \;+ \;\sqrt{46}}{12},$ which is approximately $0.73186.$ For a lot more about the Besicovitch $\frac{1}{2}$-problem, see Hany M. Farag's papers at

http://www.math.caltech.edu/people/farag.html

(50 minutes later)

Here's more about the first Besicovitch reference:

On p. 454 of the paper below, about $\frac{2}{3}$ down the page, is the following (italics in the original). [Note: The paper is freely available on the Internet.]

At almost all points of an irregular set the lower density is less than $1 - 10^{-2576}.$

Abram Samoilovitch Besicovitch (1891-1970), On the fundamental geometrical properties of linearly measurable plane sets of points, Mathematische Annalen 98 (1928), 422-464.

The article "Strange Series and High Precision Fraud", J. M. Borwein and P. B. Borwein, The American Mathematical Monthly, Vol. 99, No. 7, (Aug-Sep, 1992), pp. 622-640, contains some interesting small error terms. In particular on p.639: $$\left|\sqrt\pi-\left(\frac{1}{10^5}\sum_{n=-\infty}^{\infty}e^{-n^2/10^{10}}\right)\right|\leq 10^{-4.2\cdot 10^{10}} $$