Economics is not an exact science and so we don't have Physical Constants i.e. quantities with a universal fixed numerical value.
That said, there are many concepts in economics that predominantly appear as constants in economic models (although in principle they can vary between individuals or across time, and there exists research that explores what happen when they do vary).
Well-known examples include the depreciation rate of capital (usually $\delta$), population growth rate (usually $n$), the rate of pure-time preference (usually $\rho$).
In calibration studies (either purely simulated or combined with empirical data), such constants tend to be assigned widely used "benchmark values" (i.e. quasi- "fixed values") that come from empirical experience usually supported over some decades of data and many countries. For example when calibrated and not estimated we often see $\delta =0.05$, $n=0.01$, $\rho=0.02$.
On another front, Euler's $e$, the base of the natural logarithms, emerges naturally when calculating compound interest rate continuously (and not in discrete intervals). Assume we make a deposit $D$ at yearly interest rate $r$. If interest is calculated once per year, at the end of one year we will get $D(1+r)$. Assume now that we calculate interest monthly, add the interest to principal and then next month calculate interest again. After one year we will get
$$D\left (1+ \frac {r}{12}\right)^{12}$$
In general if we partition the year in $n$ periods we will have
$$D\left (1+ \frac {r}{n}\right)^{n}$$
and if we let $n$ go to infinity we get
$$\lim_{n\to \infty}\left (1+ \frac {r}{n}\right)^{n} = e^r$$
Finally, there are certain "stylized facts" , according to which various economic ratios appear to be "stable" (if not strictly "constant") over time, like the income share of production inputs, or the capital/output ratio. The seminal reference here is
Kaldor, Nicholas, “Capital Accumulation and Economic Growth,” in F.A. Lutz and D.C. Hague, eds., The Theory of Capital, St.Martins Press, 1961, pp. 177–222.
while see
Jones, C. I., & Romer, P. M. (2010). The new Kaldor facts: Ideas, institutions, population, and human capital. American Economic Journal: Macroeconomics, 2(1), 224-245.
for extensions and updates.
