How can I get the user not to be listed twice?
Try the SQL below. I could only create a few sample records, so maybe I haven't covered everything. 

BTW, welcome to the forum. In future, when you are asking questions like this one, it would be great if you could provide DDL in the form of CREATE TABLE MyTable(col1, col1_type, col2...) and some sample DML (INSERT INTO TABLE MyTABLE VALUES(val1, val2...). You can put this up on SQLFiddle for example - or here of course if there's not too much data - you can always provide a downloadable script. 

Good questions tend to get better answers - check out this (and links within) to see how to get the best out of this forum - it really is a great resource. As it says, help us to help you!

SELECT
    *
FROM
    racerecords
WHERE
    (holder_regid, rec_time) IN (
        SELECT
            holder_regid,
            MIN(rec_time)
        FROM
            racerecords
        WHERE
            race_id = 17
        GROUP BY
            holder_regid
    )
AND race_id = 17 
GROUP BY holder_regid   &lt;&lt;=== added this line
ORDER BY rec_time, holder_regid;