"INSERT INTO times (`times_id`,`times_raceid`, ...)
VALUES ('','$a_raceid1' , ...) , ('','$a_raceid2' , ...) , ..."

In your case, you could ask for how many records will be added, like you said, then enumerate the form field names:

record1:
...name=a_meet_id1
...

record2:
...name=a_meet_id2
...

record3:
...name=a_meet_id3

Then use...
$a_raceid1 = $_POST['a_meet_id1'];
$a_raceid2 = $_POST['a_meet_id2'];
$a_raceid3 = $_POST['a_meet_id3'];

along with your new multiple INSERT INTO statement (also dynamically created).