Edit: Given OP's comment on my answer, I think the answer may be
Space (or perhaps Outer Space)
A lot of the reasoning remains the same.
It would be even above what a bird could reach so may be what they dream of
For as long as I can remember, I've had this dream.
Let me reassure you, it's a dream many have seen.
Many people have dreamed of becoming astronauts
I rise above this terrestrial sphere onto a higher plane.
Given this incredible power, I break out of my chains.
Reaching into outer space would break free from the atmosphere (terrestrial sphere)
How far will I be able to travel? Only my mind limits me.
Space is very large, possibly infinite.
To have bestowed upon me this incredible gift, a hero I'll be.
Astronauts are lauded as heroes, think the moon landing.
Also Rocketman, the famous Elton John song.