Solution:
Answer: 359108

We count the complement - the number of functions $f$ such that for all $i \in A$, $\left|f(i)-f^{-1}(i)\right| \leq 1$.

The condition is equivalent to $|f(f(i))-i| \leq 1$ for all $i \in A$. If $f(j)=j$, the inequality is automatically satisfied for $i=j$. Otherwise, if $f(f(j))=j$ but $f(j)=k \neq j$, then we will have $f(f(k))=k$, allowing the inequality to be satisfied for $i=j$, $k$. Else, if $f(f(i)) \neq i$, say $f(f(i))=i+1$ and $f(i)=k$, then $f(f(k))=f(i+1)=k+1$ or $k-1$. Thus the function $f$ allows us to partition the elements of $A$ into three groups:
(a) those such that $f(i)=i$,
(b) those that form pairs $\{i, j\}$ such that $f(i)=j$ and $f(j)=i$, and
(c) those that form quartets $\{i, i+1, j, j+1\}$ such that $f$ permutes them as ( $\begin{aligned} & i \\ & j\end{aligned} i+1 \quad j+1$ ) or (i $j+1 \quad i+1 \quad j$ ), in cycle notation.

Let $a$ be the number of elements of the second type. Note that $a$ is even.

Case 1: There are no elements of the third type. If $a=8$, there are $9 \cdot 7 \cdot 5 \cdot 3=945$ possibilities. If $a=6$, there are $\binom{9}{3} \cdot 5 \cdot 3=1260$ possibilities. If $a=4$, there are $\binom{9}{5} \cdot 3=378$ possibilities. If $a=2$, there are $\binom{9}{7}=36$ possibilities. If $a=0$, there is 1 possibility. In total, case 1 offers $945+1260+378+36+1=2620$ possibilities.

Case 2: There are 4 elements of the third type. There are 21 ways to choose the quartet $\{i, i+1, j, j+1\}$. For each way, there are two ways to assign the values of the function to each element (as described above). For the remaining 5 elements, we divide into cases according to the value of $a$. If $a=4$, there are $5 \times 3=15$ possibilities. If $a=2$, there are $\binom{5}{3}=10$ possibilities. If $a=0$, there is one possibility. In total, case 2 offers $21 \times 2 \times(15+10+1)=1092$ possibilities.

Case 3: There are 8 elements of the third type. There are 5 ways to choose the unique element not of the third type. Of the remaining eight, there are 3 ways to divide them into two quartets, and for each quartet, there are 2 ways to assign values of $f$. In total, case 3 offers $5 \times 3 \times 2^{2}=60$ possibilities.

Therefore, the number of functions $f: A \rightarrow A$ such that for at least one $i \in A$, $\left|f(i)-f^{-1}(i)\right|>1$ is $9! -2620-1092-60=359108$.