Recursive algorithm - a^(2^n)
Let's look at how many multiplications are necessary to perform $a^{16}$ by your recursive method:

$$a \xrightarrow{SQR} a^2 \xrightarrow{MUL} a^3  \xrightarrow{MUL} \ldots  \xrightarrow{MUL} a^{14}  \xrightarrow{MUL} a^{15}  \xrightarrow{MUL} a^{16}$$

Where: 

$SQR$ = Squares a number

$MUL$ = Usual Multiplication

So as Yuval pointed out in the comments, even a small power of $2$ would require $\Theta(n)$ multiplications, which means this would be too slow for practical use. 

However, you can the same thing using only $SQR$ with $\Theta(\log{n})$ operations. That is:

$$a \xrightarrow{SQR} a^2 \xrightarrow{SQR} a^4 \xrightarrow{SQR} a^8 \xrightarrow{SQR} a^{16}$$

Of course, since $16$ is a power of $2$ we only need to use $SQR$. But you can alternate between $SQR$ and $MUL$. This is the principle behind Exponentiation by Squaring.

Example:Computing $a^{19}$.

$$a \xrightarrow{SQR} a^2 \xrightarrow{SQR} a^4 \xrightarrow{SQR} a^8 \xrightarrow{MUL} a^{9} \xrightarrow{SQR} a^{18} \xrightarrow{MUL} a^{19} $$