Peeyaamoniaded-zeroed is defined as a number smaller than 0 equal to 0.3^3^33^33^333^33^3^0.3. This number however would be greater than 0, an extremely small positive real number, so it can't be smaller than 0. There is no known resolution for this problem.
Computation[]
We can compute the value of this expression using standard mathematics, by beginning at the top of the power tower and working our way down. This is a perfectly valid expression, and in fact we can evaluate any power tower where all of it's terms are in the real interval (0,infinity).
So we first have 0.3, then 3^0.3 which is equal to about 1.39. Then we compute 33^3^0.3 as 33^1.39 which is about 129.04. Next we have 333^129.04 which we can approximate as 10^(129.04 * log(333)) which works out to about 10^325.5. Next we evaluate 33^10^325.5, which would be 10^(10^325.5 * log(33)) which would be 10^10^(325.5 + log(log(33))) bringing us to 10^10^325.69. This number is now larger than a googolplex and already larger than 10^300,000,000,000, which by comparison is only 10^10^11.48. Next we evaluate 33^10^10^325.69. This becomes 10^(10^10^325.69 * log(33)) = 10^10^(10^325.69 + log(log(33))). log(log(33)) is only 0.18, and so it's effect to 10^325.69 can be ignored. We therefore conclude this is still equal to about 10^10^10^325.69. Next we evaluate 3^10^10^10^325.69, however this would only amount to about halving the exponent 10^10^10^325.69 which would appear negligible at this scale. So we can safely conclude this is about 10^10^10^10^325.69. Lastly we have 0.3^10^10^10^10^325.69. Since 0.3 is less than 1 however we can take the log to obtain -0.52. So we get 10^(-0.52 * 10^10^10^10^325.69). As before the multiplying by about 1/2 has no noticable effect on the power tower, but this has the effect of turning the whole exponent negative, making it an extremely small number instead of an extremely large number. In fact we can approximate it as 10^(-10^10^10^10^325.69), which can be written as 1/10^10^10^10^10^325.69, which can be written more succintly as 1/(E325.69#5). As inconceivably tiny as this value would be it would still be greater than 0 (in every sense of the word) and would still not be infinitesimal. In fact, this number would be allowed in googology.