What does $2^π$ : the multiplication of $2 \pi$ many times? This is rather an intuitive question, in the sense that indeed a real number raised to the power of an irrational doesn't make sense but I wanted to know what does it mean intuitively. If $2^7=2*2*2*2*2*2*2, 2^{\frac{1}{2}}=\frac{1}{\sqrt{2}}$ and $a^b=a*a*a*a*a*...*a\, \,  b$ many times, I'm curious as to what does $2^π$ mean. Obviously it's equal to $8.8249778271...$ but what's the meaning in the aforementioned scenario?

Start with $2^3=8$.  Then think about what $2^{3.1}$ is:
$$2^{3.1} = 2^{31/10} = \sqrt[10]{2^{31}} \approx 8.574187700.$$
Then consider 
$$2^{3.14} = 2^{314/100} = \sqrt[100]{2^{314}} \approx 8.815240927.$$
Since the sequence $3, 3.1, 3.14, 3.141, \ldots$ approaches $\pi,$, the
sequence
$$2^3, 2^{3.1}, 2^{3.14}, 2^{3.141}, \ldots$$
should approach $2^\pi.$  So it's probably best to think of $2^\pi$ as a limit.
