If we look at circular low Earth orbits only:
height speed period
km m/s hours:min:sec
200 7789.1 1:28:21
300 7730.5 1:30:22
400 7673.2 1:32:24
500 7617.2 1:34:28
600 7562.3 1:36:32
700 7508.7 1:38:37
800 7456.1 1:40:43
900 7404.7 1:42:50
1000 7354.3 1:44:21
The lowest orbit has the fastest speed. But below 400 km orbits decay very fast, 300 km within 6 month, 200 km in about a day.
Now we look at elliptical orbits:
min at min max at max
height speed height speed period
km m/s km m/s hours:min:sec
400 7701.3 500 7589.2 1:33:26
400 7728.9 600 7507.1 1:34:28
400 7755.9 700 7426.9 1:35:30
400 7782.5 800 7348.4 1:36:32
400 7834.3 1000 7196.6 1:38:37
400 9127.0 10000 3774.9 3:26:26
400 10521.9 100000 669.8 37:11:36
400 10677.8 200000 350.3 96:10:06
400 10762.3 400000 179.3 259:31:25
So a very elliptical orbit has the fastest speed, but only when close to Earth at minimal height. But the period gets much longer and the average speed is lower.The last line is an elliptical orbit to the moon and back. This speed record is held by the Apollo missions. (For simplicity, the orbit was calculated without the influence of the Moon.)
All orbits were calculated using this webpage by Bernd Leitenberger. It is only available in German.