After 6 years I still don't understand this

What you're saying is that the rate of change of the bacteria is proportional to its current population (that's how you get exponential growth). Therefore, you claim:

dx/dt = kx

dx/dt -kx = 0

(dx/dt)(e^(-kt)) +x*(-k*e^(-kt)) = 0

∫ (dx/dt)(e^(-kt)) +x*(-k*e^(-kt)) dt = C

x*(e^(-kt)) = C

x = C*e^(kt)

x = C*(e^k)^t

Since you specified that the bacteria is doubling in population every unit of time, e^k = 2. Therefore, you have:

x = C*2^t

You also specified that the initial condition that there is one unit of bacterial population. Therefore:

1 = C*2^(0) = C

C = 1

Thus, your model states:

x = 1*2^t

If your initial condition is 0, then:

x = C*2^t

0 = C*2^(0) = C

C = 0

x = 0*2^t = 0

Obviously, in an initial population of 0, there will be nothing going on, because doubling 0 will result in 0 and so on. You haven't proven anything at all regarding 0^0.

The 0^0 argument is like the why .999... = 1 argument (sort of). In that people tend to argue craploads for multiple sides and no one like to agree with the other side. 0^x = 0 for any nonzero, thus people say that 0 ^ 0 should be 0. But x ^ 0 = 1 for any non zero x, thus people say that 0^0 should be 1.

Actually, the .999... = 1 argument is completely different from the 0^0 argument. By using summations, fractions, and various other methods, there actually exists valid proof that .999... = 1. For 0^0=0, all arguments supporting 0^0=0 are incomplete, including the 0^x argument.