why is $2.2250738585072014\text{e}{-308}$ not a number? In programming the min value of a float is: $$2.2250738585072014\text{e}{-308}$$

but when I type this into a calculator, it says Not a Number. what I am wondering is why this is an invalid number?

Note: I am only just finishing grade 8 math and haven't learned fully what the 'e' in an equation does. I think it has something to do with putting the number to the power of ten times the number after the e.
You are correct about your interpretation of the $e$ stuff. Indeed, this value is 

$2.2250738585072014 \cdot 10^{-308}$

Which is, to be sure, a number. It's just that, in order to properly store this number, the calculator will need a fair bit of memory to store it accurately, and you've essentially attempted to exhaust it.
In the so called "scientific notation" of numbers the e stands for 10 to the power of x. So in your case 2.22... times 10 to the power of -308.
Your number happens to be the smallest number representable by the very common data type "double", a 64 bit binary floating point representation. So your calculator is unable to represent it because it is too small and wrongfully tells you it was not a number when in fact it is a number, simply too small.