Alice, 1311 bytes
w.2~E/ O \hK\h
The trailing newline is required
w K Main loop
. Duplicates the top of the stack, or two `0` if the stack is empty
2~ 2~E Pushes two on the stack, swap the top two items
and calculate E Calculates `2^n`, where `n` is the current iteration minus 1
/ O \ Prints the number
h Increments the exponent