With dplyr that would be:
df %>%
group_by(country, idx = cumsum(event == 1L)) %>%
mutate(counter = row_number()) %>%
ungroup %>%
select(-idx)
#Source: local data frame [10 x 4]
#
# country year event counter
#1 A 2000 0 1
#2 A 2001 0 2
#3 A 2002 1 1
#4 A 2003 0 2
#5 A 2004 0 3
#6 B 2000 1 1
#7 B 2001 0 2
#8 B 2002 0 3
#9 B 2003 1 1
#10 B 2004 0 2
Or using data.table:
library(data.table)
setDT(df)[, counter := seq_len(.N), by = list(country, cumsum(event == 1L))]
Edit: group_by(country, idx = cumsum(event == 1L)) is used to group by country and a new grouping index "idx". The event == 1L part creates a logical index telling us whether the column "event" is an integer 1 or not (TRUE/FALSE). Then, cumsum(...) sums up starting from 0 for the first 2 rows, 1 for the next 3, 2 for the next 3 and so on. We use this new column (+ country) to group the data as needed. You can check it out if you remove the last to pipe-parts in the dplyr code.