Get dplyr count of distinct in a readable way

How about this option:

data %>%                    # take the data.frame "data"
  filter(!is.na(aa)) %>%    # Using "data", filter out all rows with NAs in aa 
  group_by(bb) %>%          # Then, with the filtered data, group it by "bb"
  summarise(Unique_Elements = n_distinct(aa))   # Now summarise with unique elements per group

#Source: local data frame [3 x 2]
#
#  bb Unique_Elements
#1  a               2
#2  b               1
#3  c               1

Use filter to filter out any rows where aa has NAs, then group the data by column bb and then summarise by counting the number of unique elements of column aa by group of bb.

As you can see I'm making use of the pipe operator %>% which you can use to "pipe" or "chain" commands together when using dplyr. This helps you write easily readable code because it's more natural, e.g. you write code from left to write and top to bottom and not deeply nested from inside out (as in your example code).

Edit:

In the first part of your question, you wrote:

I know I can do things like:

by_bb<-group_by(data, bb, add = TRUE)
summarise(by_bb, mean(aa, na.rm=TRUE), max(aa), sum(!is.na(aa)), length(aa))

Here's another option to do that (applying a number of functions to the same column(s)):

data %>%
  filter(!is.na(aa)) %>%
  group_by(bb) %>%
  summarise_each(funs(mean, max, sum, n_distinct), aa)

#Source: local data frame [3 x 5]
#
#  bb mean max sum n_distinct
#1  a    2   3   4          2
#2  b    2   2   2          1
#3  c    4   4   4          1