I'm sure the list you seek would be almost endless. One may suggest that you browse the books by Bruce C. Berndt, *Ramanujan's Notebooks*, Part I, II, etc, Springer.

Euler's formula $e^{\pi i}+1=0$ is everyone's favorite. In the same spirit, but to show the massive computational power of Ramanujan, here is special case from Entry 17, page 435, Part V, of the above-mentioned series.
$$\sum_{k=1}^{\infty}\frac{(-1)^{k-1}k}{e^{k\pi\sqrt{3}}-(-1)^k}=\frac1{4\pi\sqrt{3}}-\frac1{24}.$$
Notice the interplay of the two famous constants $\pi$ and $e$.

In view of Robert Israel's reasonable comment, perhaps we could go for the modest expressions:
$$\sqrt{2\left(1-\frac1{3^2}\right)\left(1-\frac1{7^2}\right)\left(1-\frac1{11^2}\right)\left(1-\frac1{19^2}\right)}
=\left(1+\frac17\right)\left(1+\frac1{11}\right)\left(1-\frac1{19}\right)$$
found in S. Ramanujan, *Notebooks of Srinivasa Ramanujan*, Volume II,
Tata Institute of Fundamental Research, Bombay, 1957. See pp. 309 and 363.