The reported value is
$$\hat\gamma_2^{(2)} = \frac{n(n+1)}{(n-1)(n-2)(n-3)}\sum_{i=1}^{n}\left(\frac{x_i - \bar{x}}{s}\right)^4 - \frac{3(n-1)^2}{(n-2)(n-3)},$$
where \(s\) is the (divisor-\(n-1\)) sample standard deviation.
Subtracting the asymptotic correction \(3(n-1)^2/((n-2)(n-3))\)
centers the statistic at 0 for a normal distribution; that is, excess
kurtosis is reported (rather than “raw” kurtosis, which centers
at 3).
Why isn't this in base R? See the same Details in
skewness: R Core defers higher-order moment statistics to
contributed packages, partly because multiple formulas (biased Type 1,
bias-corrected Type 2, Minitab Type 3) coexist. DMAR adopts Type 2,
the form most common in psychometric reporting and used internally by
descriptives.
Diagnostic interpretation. As a rough rule of thumb,
\(|\mathrm{kurtosis}| > 7\) is sometimes flagged as indicative of
departures from normality large enough to threaten normal-theory
inference (e.g., maximum likelihood estimation in factor analysis or
structural equation modeling).