You must defeat either Athena or Red Arremer to see the real endings. Note you can save your progress at the last stage and such.
You can fight Athena for your ninth opponent if:
- You don't lose any rounds for stages 1 thru 7.
- You don't win any round via block damage
(this is when you hit an opponent with a special move or S^ move,
and they block it but still take a little bit of damage as a result).
You can fight Red Arremer for your ninth opponent if:
- You don't lose any rounds for stages 1 thru 7.
- You win at least 5 rounds with a special move (winning by block
damage is okay).
Otherwise, the game ends after the 8th stage. Obviously, you can do
whatever you like during the eighth stage and it won't affect the chance
of meeting Athena or Red Arrmer.
If you do lose a round during stages 1 thru 7, just lose completely
and continue, and you can still meet the Secret Bosses (since the key
is to not lose any _rounds_. It doesn't matter if you lose the whole
battle and continue).
Finally the number of continues and current game time have nothing to
do with fighting these characters.