What are some large or small mathematical constants? I understand this question is a bit vague, but I would like to know about notable mathematical constants that are large or small, and I clarify what I mean now:


*

*Notable, as in not a product, sum, exponentiation, or other operation on other constants simply for the sake that they are a product, sum, exponentiation, or operation on other constants. So, not $e^{e^2}$ or $\pi^{50\gamma}$. Also, not large or small for the sake of being large or small (no googolplex.)

*Large or small, as in larger than 100 or less than 0.1.


Motivation: there is one number on Wikipedia's list of mathematical constants page larger than 100, and three less than 0.1, and I am stunned by this normality! We need better representation of large and small constants.
 A: 
  
*
  
*Constant $\sigma_3$: The non-zero constant with smallest absolute  value stated in Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94 is 
  \begin{align*}
\color{blue}{\sigma_3=-0.000\,111\,158\,2...}
\end{align*}

We can find in section 2.21 Stieltjes Constants:
\begin{align*}
\sigma_n=\sum_{\rho}\frac{1}{\rho^n}=
\begin{cases}
-\frac{1}{2}\ln(4\pi)+\frac{\gamma_0}{2}+1=0.023\,095\,708\,9\ldots&n=1,\\
-\frac{\pi^2}{8}+\gamma_0^2+2\gamma_1+1=-0.046\,154\,317\,2\ldots&n=2,\\
-\frac{7\zeta{3}}{8}+\gamma_0^3+3\gamma_0\gamma_1+\frac{3\gamma_2}{2}+1=\color{blue}{-0.000\,111\,158\,2\ldots}&n=3,
\end{cases}
\end{align*}
where    each sum is over all nontrivial zeros $\rho$ of the Riemann zeta function $\zeta(z)$. The constants $\gamma_n$ come from the Laurent expansion in a neighborhood of its simple pole at $z=1$:
\begin{align*}
\zeta(z)=\frac{1}{z-1}+\sum_{n=0}^\infty \frac{(-1)^n}{n!}\gamma_n(z-1)^n.
\end{align*}

  
*
  
*Stieltjes Constant $\gamma_{100\,000}$:
The coefficients $\gamma_n$ can be proved to satisfy
  \begin{align*}
\gamma_n=\lim_{m\to\infty}\left(\sum_{k=1}^m\frac{\ln(k)^n}{k}-\frac{\ln(m)^{n+1}}{n+1}\right)
\end{align*}
  In particular $\gamma_0=\gamma=0.577\,215\,664\,9\ldots$ is the Euler-Mascheroni constant. A somewhat larger constant of this family is the constant
  \begin{align*}
\color{blue}{\gamma_{100\,000}=1.991\,927\,306\,3\ldots\times10^{83\,432}}
\end{align*}
  which is stated here.

A: I'll start. Ramanujan's constant (sometimes called the Hermite-Ramanujan constant) is a large transcendental number that is very close to an integer. It is expressible as
$$R = e^{\pi\sqrt{163}} = 262537412640768743.99999999999925\dots \approx 640320^3+744$$
A: Some are very arbitrary, like $\pi^\pi$.
If that number is considered important enough for inclusion in the wiki list, then why not $\pi^{\pi^{\pi}}$, $e^e$, $e^{e^e}$, and many other tetrations?
A: As to why notable constants are generally small, I think it is best to think about what they mean and where they come from.
For example, pi is the ratio between a circles radius and its circumference. These are 2 relatively "similar" (as in the ratio between the 2 is relatively small) distances, so pi is a relatively small number.
Gamma is approximately the difference between the harmonic series in respect to n and ln(n). Both ln and the harmonic series grow very slowly and close together (about a gamma distance apart!) but both diverge. 
Similar philosophies can be applied to most commonly used numbers like these. 
A: Wyler's constant is defined as 
\begin{align}\alpha_{\small{W}}&=\frac{9}{8\pi^4}\left(\frac{\pi^5}{2^45!}\right)^{1/4}\\&=0.0072973481300\dots\\&=
\frac{1}{137.0360824\dots}
\end{align}
(Wyler 1969; OEIS A180872), which at the time it was proposed, agreed with experiment to within $\pm1.5$ ppm for the value of the fine structure constant $\alpha$ in physics. 
The constant is a transcendental number.
