My friend think 10:08 is the all-time brightness champion. Does this seem correct to you?
I just looked at the number of lines needed to produce each number on my clock and the valid times and your friend appears to be correct.
However, “all time champion” status usually implies some type of unlimited contest. Tell your friend that he is full of it.
21:08 easily beats it on a military time digital clock (also in use in many countries as the standard format).
Yes, excepting the 88:88 of an unset clock.
PM, with the alarm on, for those two dots.
How about 22:08? 
Don’t know about that; 10:08 uses 21 segments, and 21:08 uses 20.
23:08, the apparent max, uses 23 segments.
I was counting these up quickly; can anyone correct or beat these?
Yes, 23 segments. Plus colon , plus two dots (as described above), plus the icon that indicates the radio alarm setting.
20:08 - 24 segments
00:08 - 25 segments
08:08 - 26 Segments
I thought about that one and 08:08, too, but didn’t list it because I thought my clocks display 0:08 and 8:08, respectively. Now that I checked, it seems they do use all four digits.
What about a clock with a seconds display 
08:08:08