The basic fix is to replace the fake verb 'to or', conjugated not really properly as "or's" or "or'ing", with the real noun 'bitwise-or'.

help(doctest.testmod)
...
    Optional keyword arg "optionflags" or's together module constants,
    and defaults to 0.  This is new in 2.3.  Possible values (see the
    docs for details):

Eliminate 'this is new...' here and above. Suggested replacement:

    Optional keyword arg "optionflags" (default 0) is the bitwise-or
    of the following module constants (see the docs for details):

help(doctest.testfile)
...
    Optional keyword arg "optionflags" or's together module constants,
    and defaults to 0.  Possible values (see the docs for details):

Same replacement (whatever we decide on).

Lib 25.2.4. Basic API
doctest.testfile entry:
...
Optional argument optionflags or’s together option flags. See section Option Flags and Directives.

->
Optional argument optionflags is the bitwise-or of options flags. See section Option Flags and Directives.

doctest.testmode entry refers back to above.

25.2.3.5. Option Flags and Directives

doctest.COMPARISON_FLAGS 
A bitmask or’ing together all the comparison flags above.
doctest.REPORTING_FLAGS 
A bitmask or’ing together all the reporting flags above.

A bitmask that is the bitwise-or of all the comparison flags above.
ditto with 'reporting' instead.

-or-
A bitmask -- the bitwise-or of ...

I think this is all. Searching on "or'" does not work because sphinx nicely replaces "'" with a non-ascii unicode char ;-).