More generally, I think you might want to get "top" of the rows that are sorted within a given group.
For the case of where a single value is max'd out, you have essentially sorted by only one column. However, it's often useful to hierarchically sort by multiple columns (for example: a date column and a time-of-day column).
# Answering the question of getting row with max "value".
df %>%
# Within each grouping of A and B values.
group_by( A, B) %>%
# Sort rows in descending order by "value" column.
arrange( desc(value) ) %>%
# Pick the top 1 value
slice(1) %>%
# Remember to ungroup in case you want to do further work without grouping.
ungroup()
# Answering an extension of the question of
# getting row with the max value of the lowest "C".
df %>%
# Within each grouping of A and B values.
group_by( A, B) %>%
# Sort rows in ascending order by C, and then within that by
# descending order by "value" column.
arrange( C, desc(value) ) %>%
# Pick the one top row based on the sort
slice(1) %>%
# Remember to ungroup in case you want to do further work without grouping.
ungroup()