Question

In crontab, can you do something like this?

* * * * * echo $( date +%F) >> /path/date.txt

Answer 1

The actual problem of your crontab line is not the $() or the backquotes. The problem is the percent sign %. It has a special meaning in crontabs.

From the manpage:

...
Percent-signs (%) in the command, unless escaped with backslash (\), 
will be changed into newline characters, and all data after the 
first % will be sent to the command  as standard input.
...

If you escape the percent sign with \ it should work as expected:

* * * * * echo $(date +\%F) >> /tmp/date.txt

or

* * * * * echo `date +\%F` >> /tmp/date2.txt

both work on my site.