'nl *'
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nl - the line numbering utility takes one or more file names on its command line. So nl * cats the contents of all the files with line numers.

46$ ls | wc
      15      15     233

47$ nl * | wc
     857    3540   29899

48$ ls | nl
     1  Makefile
     2  RCS
     3  Tests
     4 
     ...
    15
[download]

But you just want the count, not the list, so you would need to | tail -1 | cut -d' ' -f1.

Or to be precise: ls | nl | tr -s ' ' ' ' | tr "\t" ' ' | cut -d' ' -f2

--
TTTATCGGTCGTTATATAGATGTTTGCA

Hm, I think with this glob the shell will first try to put all filenames on your commandline. If you have a lot of files, the maximum size for a command line will be exceeded (this maximum can be pretty large on *nix machines, but the OP intended to count lots of files...).