How can I get the maximum and minimum values after applying LOESS

The ggplot uses the loess function of base R. You can run the setting outside the graph command and get the values using predict :

set.seed(4)
dados <- data.frame(date_time = 1:20, duracao = rnorm(20))
dados.loess <- loess(duracao ~ date_time, data = dados)

summary(predict(dados.loess))
#    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
#-0.06347  0.15680  0.34110  0.35310  0.56030  0.74970

Confirming the result, we can add the rows in the chart:

library(ggplot2)
ggplot(data = dados, aes (x = date_time, y = duracao)) +
  geom_point() + stat_smooth(method = "loess") +
  geom_hline(aes(yintercept = min(predict(dados.loess)))) +
  geom_hline(aes(yintercept = max(predict(dados.loess))))

When you use the stat_smooth function of ggplot2 , it performs the following steps:

I will use the mtcars database, but you can easily replace it with your own.

modelo <- loess(mpg ~ hp, data = mtcars) # ajusta o modelo  loess
mtcars$pred <- predict(modelo) # calcula as predicoes

ggplot (data = mtcars, aes (x = hp, y = mpg)) + geom_point() + 
 geom_line(aes(y = pred), colour = "blue") # plota as predicoes usando linha

See that this chart is the same as the graph made using:

ggplot (data = mtcars, aes (x = hp, y = mpg)) + geom_point() + 
 stat_smooth(method = "loess")

So, to get the maximum value of the curve just filter in your data.frame:

mtcars[mtcars$pred == max(mtcars$pred), ]