How to write Euler's e with its special font

I think this is simply an "e" in a Times clone. Your image seems to have been produced with mathptmx.

\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage{mathptmx}
\begin{document}
\[
e \equiv \lim_{n\to\infty} \biggl( 1 + \frac{1}{n} \biggr)^n = 2.7182818284590452353602874713526624977572\hdots
\]
\end{document}

Note however that mathptmx is obsolete; if you want to use a Times clone you should use newtx instead.

I’m going to give some alternatives, if you don’t mind. You already got a good answer to the question of which e that is.

There is a codepoint for “Euler’s constant” defined in Unicode, ℇ (U+2107), which is \Eulerconst in some packages. It’s not widely-used, and the consensus is that most fonts’ glyphs for it are ugly.

On the other hand, \mathrm{e} is nearly always identical to e from the plain text font. This conforms to the convention of typesetting constants upright and variables in italic. (Before egreg shows up here to point it out, I will mention that not all mathematicians agree with that.) It would be possible for an OpenType math font to use its ASCII codepoints for upright math symbols that are not copied from the text font, or for a legacy math font to supply a \mathrm alphabet different from the text font, but as of 2020, no math fonts do so.

So I personally like to patch in upright math constants from a different upright font. One of my favorite combinations is Hermann Zapf’s Palatino and AMS Euler. It’s not the style you were asking about, but here is a sample of Euler’s identities, typeset with free clones of Euler and Palatino. It requires LuaLaTeX or XeLaTeX, and the OpenType font Neo Euler.

\documentclass[varwidth, preview]{standalone}

\usepackage{mathtools}
\usepackage{unicode-math}

\defaultfontfeatures{Scale=MatchLowercase}
\setmainfont{TeX Gyre Pagella}[
  Scale=1.0,
  Ligatures={Common, Discretionary, TeX}]
\setmathfont{Asana Math}
\setmathfont[range={up/{Latin,latin,Greek,greek},
                    bfup/{Latin,latin,Greek,greek},
                    cal, bfcal, frak, bffrak},
              script-features={},
              sscript-features={}
            ]{Neo Euler}

\newcommand\upi{\symup{i}}
\newcommand\upe{\symup{e}}

\begin{document}
\begin{align*}
  \upe^{\upi x} &= \cos{x} + \upi \sin{x} \\
  \upe^{\upi \uppi} + 1 &= 0
\end{align*}
\end{document}

If you need a matching sans-serif font, you can use Zapf’s Optima (or its clone URW Classico, on CTAN but with a restrictive license). An excellent Humanist monospace font is Inconsolata.

You can substitute basically any fonts of your choice here. As another example, an upright companion for Computer Modern might be

\usepackage[regular]{newcomputermodern}

\setmathfont{CMU Serif Upright Italic}[
  range=up/{Latin,latin,Greek,greek},
  Scale=MatchLowercase ]

It is also possible to do something similar in legacy PDFTeX, but not nearly as standardized. Here is the Euler and Palatino example with the Type 1 versions of the fonts:

\documentclass[varwidth, preview]{standalone}

\usepackage{mathtools}
\usepackage{tgpagella, newpxmath}
\usepackage{bm}

\DeclareSymbolFont{euler}{U}{zeur}{m}{n}
\SetSymbolFont{euler}{bold}{U}{zeur}{b}{n}

\DeclareMathSymbol{\uppi}{\mathalpha}{euler}{"19}
\DeclareMathSymbol{\upe}{\mathalpha}{euler}{"65}
\DeclareMathSymbol{\upi}{\mathalpha}{euler}{"69}

\begin{document}
\begin{align*}
  \upe^{\upi x} &= \cos{x} + \upi \sin{x} \\
  \upe^{\upi \uppi} + 1 &= 0
\end{align*}
\end{document}

Matching an upright font to other math fonts would need to be done on a case-by-case basis, and quickly gets complicated. For Computer Modern, you might take the Latin letters from the upright italic style, {T1}{cmr}{m}{ui}, Greek lowercase letters from {LGR}{cmr}{m}{n}, and Greek uppercase letters from either OT1 or LGR.

Just be sure to load a Times Roman text and math font and you're all set: \textit{e}, $e$, and $\mathit{e}$ will all generate the desired glyph. If you load the newtxmath math font package, $\mathscr{e}$ may also be of interest.

\documentclass{article}
\usepackage{newtxtext,newtxmath} % Times Roman clone text and math font packages
\begin{document}
\textit{e} $\mathit{e}$ $e$ $\mathscr{e}$
\end{document}