Good question. I will list some constants groupped mainly into 4 categories: 1) hypercomplex numbers, 2) umbral calculus, 3) formal power series, infinite germs and divergent integrals and series, 4) linear operators (infinite matrices).
- Hypercomplex numbers
Split-complex (tessarine) unity $j$. Logarithm: $\ln j=\frac{i \pi }{2}-\frac{i j\pi}{2}$. Has property $i^j=ij$, where $i$ is complex unity.
Split-complex idempotents and zero divisors $1/2+j/2$ and $1/2-j/2$. Have properies $0^{1/2+j/2}=1/2-j/2$ and $0^{1/2-j/2}=1/2+j/2$
Dual unity $\varepsilon$ and its roots in higher-dimension algebras (e.g. $\varepsilon_1^2=\varepsilon$), all nilpotents. Has property $f(\varepsilon)=f(0)+\varepsilon f'(0)$
Grassmann algebra generators $\theta_i$ and their products, also all nilpotents.
Triplex numbers unities $j_1$ and $j_2$. As well as $c=i^{j_1+j_2}=\frac{1}{3} \left(-\left(\left(\sqrt{3}+1\right) j_1\right)+\left(\sqrt{3}-1\right) j_2-1\right)$ (here $i$ is usual complex unity), satisfying $c^3+c^2+c+1=0$. Logarithm: $\ln j_1=\frac{2 \pi \left(j_1-j_2\right)}{3 \sqrt{3}}$, $\ln c=\frac{1}{3} i \pi j_1-\frac{\pi j_1}{2 \sqrt{3}}+\frac{1}{3} i \pi j_2+\frac{\pi j_2}{2 \sqrt{3}}+\frac{i \pi }{3}$. $j_1^2=j_2$, $j_2^2=j_1$, $j_1j_2=1$.
- Umbral calculus
The most important is Bernoulli's umbra $B^-$ and $B^+=B^-+1$. $\operatorname{eval}B^-=-1/2$, $\operatorname{eval}(B^+)=1/2$ (here "eval" is evaluation, akin to taking real part). Logarithm of $B^-$ is undefined, but $\operatorname{eval}\ln(B^+)=-\gamma$. Links logarithms with trigonometric functions: $\operatorname{eval}\frac1{\pi }\ln \left(\frac{B^+-\frac{x}{\pi }}{B^-+\frac{x}{\pi }}\right)=\cot x$.
$e^{B^-}$ and $e^{B^+}$ respectively have evaluations $\frac 1{e-1}$ and $\frac1{1-1/e}$. $(-1)^{B^-+1/2}$ has evaluation $\frac{\pi}2$.
Euler's umbra $E$. $\operatorname{eval}E=0;$ $\operatorname{eval}\ln E=-\pi/2$
- Infinite germs and divergent integrals/series.
The most simple infinite number $\omega$ is the germ at infinity of the function $f(x)=x$. Denoted in Levi-Civita field as $\varepsilon^{-1}$ (so has inverse $\varepsilon$ in that field, as well as in Hardy field). Can be represented as divergent integrals $\int_0^\infty dx$ and $\int_0^\infty\frac1{x^2}dx$ (via Laplace transform). Half the numerocity of integers, equal to numerocity of even or odd numbers. Divergent sum representation: $1/2+\sum_{k=1}^\infty 1$. Has finite part (regularized value) $0$. Also, the germ of the function $\frac1x$ at zero (at positive direction). If we modify Delta distribution to behave like a function, $\omega=\pi\tilde{\delta}(0)$ (via Fourier transform).
$\lambda=-\int_0^1 \frac1x dx$. This is negatively infinite constant. Via its definition can be used to extend logarithmic function to zero (thus $\lambda=\ln 0$). Has finite part (regularized value) $-\gamma$. Since Harmonic series is the Riemann sum of this integral, $\lambda$ is the negative of the sum of Harmonic series: $\lambda=-\sum_{k=1}^\infty \frac1k$. From this equality (or via Laplace transform), $\int_1^\infty\frac1xdx=-\lambda-\gamma$. Thus, $\int_0^\infty \frac1x dx=-2\lambda-\gamma$. Other integral representations include: $\int_0^\infty\frac{e^{-x}-1}{x}dx,-\int_0^\infty \frac{1}{x^2+x}dx$. Can be expressed via $\omega$: $\lambda=-\ln\omega-\gamma$. Some divergent integrals can be expressed in terms of $\lambda$: $\int_0^\infty \frac{\ln t}{t} \, dt=\gamma\lambda+\gamma^2/2,$ $\int_0^{\infty } | \tan x| \, dx=-\lambda\omega$. If we generalize the notions of periods and Chow's $EL$-numbers to our extended set, then $\lambda$ would be both, because it can be represented as $\int_0^1 \frac{-1}t dt$ (integral of an algebraic function over algebraic domain) and $\ln 0$ respectively. Logarithms of zero divisors can be expressed via $\lambda$, in split-complex numbers: $\ln \left(\frac{a j}{2}+\frac{a}{2}\right)=\frac{j}{2} (\ln a-\lambda)+\frac{1}{2} (\ln a+\lambda)$, in dual numbers $\ln \varepsilon=\lambda+\varepsilon \omega$, $\varepsilon^\varepsilon=1+\varepsilon(1+\lambda)$.
$-\lambda-\gamma$. Equal to $\ln \omega$ and germ at infinity of logarithmic function $f(x)=\ln x$. Has finite part $0$. Unlike $\lambda$, is neither generalized period, nor EL-number. Has representations $\int_1^\infty\frac1xdx,$ $\int_0^\infty \frac{e^{-x}}{x}dx,$ $\int_0^\infty \frac{x-\log (x)-1}{(x-1)^2}dx$.
$\int_{-\infty}^\infty e^x dx$. Equal to $e^\omega$. Has finite part $0$. Germ of the function $f(x)=e^x$ at infinity.
- Among infinite matrices one constant is the matrix of derivative operator $D$. Many other operators can be represented via $D$: finite difference $\Delta=e^D-1$, Fourier transform $\mathcal{F}=e^{\frac{\pi i}{4}(D^2-x^2+1)}$, scaling $f(x)\to f(ax)=a^{xD}$, etc.