The question is a bit unclear - the example provided may mean you want to remove all #s, or remove the part after the last ".", or remove the part after the first "1", or even remove all charcters after character 13. Please clarify.
If you mean that you want to remove first N characters in a string (e.g. "up to a character # 13"), do echo testFile.txt.1 | cut -c14-
. To retain the chars 1-13, on the other hand, do echo testFile.txt.1 | cut -c1-13
If you mean that you want to remove the beginning characters until the first occurence of a specific character (in your example that seems to be "1"), do echo testFile.txt.1 | perl -e 's/^[^1]*//;'
. To remove everything AFTER the first "1", do echo testFile.txt.1 | perl -e 's/1.*$//;'
If you want to remove all the #s, do echo testFile.txt.1 | perl -e 's/\d//g;'
or without Perl, echo testFile.txt.1 | tr -d "[0-9]"
If you want to remove everything after the last ".", do echo testFile.txt.1 | perl -e 's/\.[^.]+/./;'