Power-tower paradox

The power-tower paradox is an apparent paradox discovered by Robert Munafo that arises in the calculation of very large numbers, usually as the result of a rounding error. The paradox is visible particularly in Hypercalc.

A (rather small) example of the paradox would be an approximation of $$50^{10^{10^{10}}}$$ using a "stack" of base-10 exponents:

$$50^{10^{10^{10}}} = 10^{\text{log}_{10}50 \cdot 10^{10^{10}}} \approx 10^{1.698970004 \cdot 10^{10^{10}}} = 10^{10^{\text{log}_{10} 1.698970004 + \text{log}_{10} 10^{10^{10}}}} = 10^{10^{\text{log}_{10} 1.698970004 + 10^{10}}} \approx 10^{10^{0.230185711 + 10^{10}}} = 10^{10^{10000000000.230185711}} = 10^{10^{1.0000000000230185711 \cdot 10^{10}}}$$

$$= 10^{10^{10^{\text{log}_{10} 1.0000000000230185711 + \text{log}_{10} 10^{10}}}} = 10^{10^{10^{\text{log}_{10} 1.0000000000230185711 + 10}}} \approx 10^{10^{10^{10.0000000000099967971146562514}}}$$

To compare this against $$10^{10^{10^{10}}}$$ would take high levels of precision and even higher levels for larger "stacks." Munafo's original example required about 100 digits of precision to compare accurately.