your attempt only opens the file for read, not write.

open RECORD_NUMBER, '<', "counter_data.dat" or die;
$count = <RECORD_NUMBER>;
close RECORD_NUMBER;
$count++
open FILE, '>', "counter_data.dat" or die;
print FILE $count;
close FILE;
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I see most people answered along these lines. It is worth mentioning open '+<' mode too, although in that case one would have to seek as well.