Mathematica, 20 bytes

As a list of '1's:

Table[1, {x, 1, 2^^ #}] & 

Anonymous function.

Mathematica, 14 bytes

As a list of '1's:

Table[1,2^^#] & 

Anonymous function. Would probably be 10 bytes in the hypothetical mthmca golfing language.

Mathematica, 33 bytes (or 22, see discussion)

StringJoin@Table["*", {x, 1, 2^^ #}] & 

Anonymous function. Remove 'StringJoin@' and subtract 11 bytes if the answer is allowed to be a list of *s rather than a string.

Mathematica, 20 bytes

As a list of '1's:

Table[1, {x, 1, 2^^ #}] & 

Anonymous function.