The minimum value of $$f(x) = a^{a^x} + a^{1 - a^x}$$, where $$a, x \in R$$ and $$a > 0$$, is equal to:

Let $$t = a^x$$, where $$t > 0$$ since $$a > 0$$. Then $$f(x) = a^t + a^{1 - t} = a^t + \frac{a}{a^t}$$.

Applying the AM-GM inequality to the two positive quantities $$a^t$$ and $$\frac{a}{a^t}$$, we get $$a^t + \frac{a}{a^t} \geq 2\sqrt{a^t \cdot \frac{a}{a^t}} = 2\sqrt{a}$$.

Equality holds when $$a^t = \frac{a}{a^t}$$, i.e., $$a^{2t} = a$$, which gives $$t = \frac{1}{2}$$. Since $$t = a^x$$, this requires $$a^x = \frac{1}{2}$$, which is achievable for a suitable real $$x$$.

Therefore, the minimum value of $$f(x)$$ is $$2\sqrt{a}$$.