Number $e$ had several "near discoveries" by Napier (1614), St. Vincent (1647) and Huygens (1661) in connection with natural logarithms before it was finally explicitly introduced... without any connection to them. Napier's was led to (natural) logarithms by considering motion of a point, whose velocity is equal to its distance to the origin (in modern notation $dx/dt=x$). St. Vincent considered areas under a rectangular hyperbola ($y=1/x$), which as Huygens observed later have the same multiplicative property as logarithms. Huygens even introduced general exponential function $ka^x$, which he called "logarithmic", and calculated what we call "logarithm to base 10 of $e$" to 17 decimal places. But he did not "see" $e$.
As MacTutor describes it, "in 1683 Jacob Bernoulli looked at the problem of compound interest and, in examining continuous compound interest, he tried to find the limit of $(1 + 1/n)^n$ as n tends to infinity. He used the binomial theorem to show that the limit had to lie between 2 and 3... if we accept this as a definition of e, it is the first time that a number was defined by a limiting process." It was also Jacob Bernoulli who started treating logarithms as functions, rather than numerical aids, and realized that they were the inverses of exponential functions (at the level of numbers Gregory might have known it before 1683). The first notation for $e$ appears in Leibniz's 1690 letter to Huygens, and it was $b$. Thus, $e$ got a proper name 16 years earlier than $\pi$ when Jones christened it.
Calculus for exponential functions first appears in Johann Bernoulli's (Jacob's brother) 1697 Principia Calculi Exponentialium seu Percurrentium, only then derivatives come into play. But only Euler pulled logarithmic, area under hyperbola, compound interest, exponential function, approximation by series, etc. threads together into a systematic treatment in his He also gave $e$ its modern name in a letter to Goldbach in 1731, not because of his own name, and possibly not even because of "exponential", but because the preceding vowel $a$ was already taken.
For more on Napier and St. Vincent see Fauvel's and Katz's articles in Learn from the Masters.