When you say "Max Race number", do you mean the number of the latest race (race with the most recent date), or the highest number?
Assuming you mean the former, and making up the name 'Races' for your table:
SELECT
Race.*
FROM
(
SELECT Jockey, MAX(Date) as MaxDate
FROM Races GROUP BY Jockey
) MaxRaceDate
inner join (
SELECT Jockey, Date, MAX(Race) as MaxRace
FROM Races
GROUP BY Jockey, Date
) MaxRaceForDate
on MaxRateDate.Jockey = MaxRaceForDate.Jockey
and MaxRateDate.MaxDate = MaxRaceForDate.Date
inner join Race
on Race.Jockey=MaxRace.Jockey
and Race.Date = MaxRace.MaxDate
and Race.Race = MaxRaceForDate.MaxRace
This query is getting long, so it's possible that I've got an error, but it should be roughly right. ;)
Basically, you need an aggregate query on the table to find the date, then another aggregate query for the maximum rate for the date, then finally the table itself to get the information. And then you need to put them together with either subqueries or joins, and joins are typically much faster, so I went that way.