R create skewed distribution using stat_function

You might be interested in the t-distribution. A t-distribution has higher kurtosis than a normal distribution. As far as I know, base R does not support skew a for t-distributions. However, the package skewt provides the option for you to generate a t-distribution with a named skew:

library(skewt)
x <- x=seq(-1,1,0.05)
y <- dskt(x=seq(-1,1,0.05), df=10, gamma=2)
plot(x, y)

The quantiles you get here are the standardised quantiles (mean is 0). You could calculate unstandardised quantiles by multiplying by the standard deviation and adding the mean (reverse standardise of sorts).

mean <- 100
sd <- 15
x1 <- x*sd + mean
plot(x1,y)

It depends what you wish to show. A normal distribution with fatter tails (larger kurtosis) is a t-distribution with n degrees of freedom. A t-distribution with different skew gamma is another layer on top.