If $x > 10$, what is the greatest possible value of the expression
\[
  {( \log x )}^{\log \log \log x} - {(\log \log x)}^{\log \log x} ? 
\]
All the logarithms are base 10.
1. Given the expression:
   \[
   {(\log x)}^{\log \log \log x} - {(\log \log x)}^{\log \log x}
   \]
   where all logarithms are base 10, and \( x > 10 \).

2. Let's assume \( x = 10^{10^{10^a}} \) for some \( a > 0 \). This assumption is valid because \( x > 10 \).

3. Calculate \( \log x \):
   \[
   \log x = \log (10^{10^{10^a}}) = 10^{10^a}
   \]

4. Calculate \( \log \log x \):
   \[
   \log \log x = \log (10^{10^a}) = 10^a
   \]

5. Calculate \( \log \log \log x \):
   \[
   \log \log \log x = \log (10^a) = a
   \]

6. Substitute these values back into the original expression:
   \[
   {(\log x)}^{\log \log \log x} - {(\log \log x)}^{\log \log x}
   \]
   becomes:
   \[
   (10^{10^a})^a - (10^a)^{10^a}
   \]

7. Simplify each term:
   \[
   (10^{10^a})^a = 10^{a \cdot 10^a}
   \]
   and
   \[
   (10^a)^{10^a} = 10^{a \cdot 10^a}
   \]

8. Therefore, the expression simplifies to:
   \[
   10^{a \cdot 10^a} - 10^{a \cdot 10^a} = 0
   \]

Conclusion:
\[
\boxed{0}
\]