0.25 \ \cdot \ 4^{3} \ - \ 1

To solve this expression, we follow the order of operations (PEMDAS/BODMAS). First, we calculate the exponent and then perform the multiplication and subtraction.

First, we have to calculate the exponent: $4^3 = 4\cdot4\cdot4 = 64$.

Now, we can substitute the value of the exponent into the expression:

$0.25 \cdot 4^{3} - 1 = 0.25 \cdot 64 - 1$

Next, we perform the multiplication: $0.25 \cdot 64 = 16$.

Finally, we subtract 1: $16 - 1 = 15$.

Therefore, $0.25 \cdot 4^{3} - 1 = 15$.