In this paper we establish some new supercongruences motivated by the well-known fact $\lim_{n\to∞}(1+1/n)^n=e$. Let $p>3$ be a prime. We prove that $∑_{k=0}^{p-1}\binom{-1/(p+1)}k^{p+1}≡ 0\ \pmod{p^5}\ \ \ \mbox{and}\ \ \ ∑_{k=0}^{p-1}\binom{1/(p-1)}k^{p-1}≡ \frac{2}{3}p^4B_{p-3}\ \pmod{p^5},$ where $B_0,B_1,B_2,…$ are Bernoulli numbers. We also show that for any $a𝟄\mathbb Z$ with $p∤ a$ we have $∑_{k=1}^{p-1}\frac1k\left(1+\frac ak\right)^k≡ -1\pmod{p}\ \ \ \mbox{and}\ \ \ ∑_{k=1}^{p-1}\frac1{k^2}\left(1+\frac ak\right)^k≡ 1+\frac 1{2a}\pmod{p}.$ ; Comment: 16 pages