To determine the maximum number of moves that Sir Alex could have made in this game, we need to analyze the operations and how each affects the game state.

Sir Alex has 9 cells initially empty. The objective is to have one cell contain the number $2^n$ at the end, while all others are empty. During the game, Sir Alex can perform the following operations:

1. Place a number $2^j$ into an empty cell.
2. Choose two cells with $2^j$ and replace one with $2^{j+1}$.

To achieve the final objective, let's consider the series of transformations necessary:

Step 1: Placing initial numbers.

- Initially, all cells are empty. During the game, numbers of the form $2^j$ (for non-negative integers $j$) are placed. Therefore, these numbers in isolation (without further operations) do not lead us directly to $2^n$.

**Step 2: Operations for obtaining $2^n$.**

Given the operations, to obtain a larger power of two from smaller powers, Sir Alex can repeatedly combine numbers. The key point is to maximize moves through combinations.

Each time Sir Alex combines two numbers $2^j$ into one $2^{j+1}$, effectively he reduces the total count of numbers by one but increases the highest power potentially.

Calculation of Moves:

1. Formulation Details:

- To achieve $2^n$ from the initial set of numbers, consider binary expansion. Achieving $2^n$ can be formulated as starting from individual units $2^0$ and combining them gradually.
- The combination of numbers proceeds from bottom levels upwards.

2. Counting

- The maximum number of $2^j$ usable is limited by the number of available cells, in this case, 9.
- It takes several steps to combine numbers, and each specific combination can be represented in a unique binomial path.

For any given $j$:

- The number of ways to initially place a total number of individual items such as $2^0$ in every possible combination is determined by $\binom{n}{i}$, where $i$ can be at most 8 (since there are only 9 cells).

- The operation count of steps for each is as you sum over these possibilities.

The formula accounts for the maximal pathways you can spawn from rearranging numbers. Removal of extra placement and merges through each summation iteratively allows a maximum number of future recombinations.

Thus, the maximum number of moves Sir Alex could have made is:
$$\boxed{2 \sum_{i=0}^{8} \binom{n}{i} - 1}$$
This formula integrates the total number of operations required based on increments over all the ways of selecting and combining up to the limits defined by the problem (9 cells).