There's 7 eggs in all. In the game, you can only carry 6 bottles. Of course, you'll need to give them to the doctor when all your bottles are full. Just don't do it in the last day, I doubt you will have enough time.
By the way, you don't need to do the alien thing again if you have Epona's song.

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They say blind arrogance sows the field of its own destruction.
<Taldarin><Tales of Symphonia spoof.><Angry Video Game Nerd>