dense_rank
it is
library(dplyr)
df %>%
group_by(section) %>%
mutate(desire=dense_rank(exhibit))
# section exhibit desired_result desire
#1 1 a 1 1
#2 1 b 2 2
#3 1 c 3 3
#4 1 c 3 3
#5 2 a 1 1
#6 2 b 2 2
#7 2 b 2 2
#8 2 c 3 3