There are a couple issues that I can see though, regarding double/triple
clicks. First, evaluate the following:

  (define-prefix-command 'test)
  (global-set-key [mouse-3] #'test)
  (global-set-key [mouse-3 mouse-1] #'forward-word)

1. Triple clicking mouse-3 results in a popup menu that seems like it
   shouldn't appear, but this is also case when triple clicking
   C-mouse-3 even without my patch. I'm not sure what it's doing there.
   
2. Double clicking mouse-3 usually results in a mouse-3 double-mouse-3
   sequence, instead of double-mouse-3. You can tell by also evaluating:

     (global-set-key [mouse-3 double-mouse-3] #'forward-line)
     (global-set-key [double-mouse-3] #'goto-line)

   With the above key set, double clicking mouse-3 calls `forward-line',
   and oddly enough _triple_ clicking mouse-3 calls `goto-line'. Here's what
   `describe-key' has to say about triple clicking mouse-3:

     <double-mouse-3> (translated from <triple-down-mouse-3>
     <triple-mouse-3>) runs the command goto-line

  I have no idea why that translation exists.

Do you have any ideas on how to solve these issues (especially getting
double click to issue double-mouse-#)? If not, maybe I'll fiddle around
with read_key_sequence later.