Sasha has $10$ cards with numbers $1, 2, 4, 8,\ldots, 512$. He writes the number $0$ on the board and invites Dima to play a game. Dima tells the integer $0 < p < 10, p$ can vary from round to round. Sasha chooses $p$ cards before which he puts a “$+$” sign, and before the other cards he puts a “$-$" sign. The obtained number is calculated and added to the number on the board. Find the greatest absolute value of the number on the board Dima can get on the board after several rounds regardless Sasha’s moves.
1. **Understanding the Problem:**
   Sasha has 10 cards with numbers \(1, 2, 4, 8, \ldots, 512\). Initially, the number on the board is 0. In each round, Dima chooses an integer \(0 < p < 10\), and Sasha chooses \(p\) cards to put a "+" sign before and the remaining \(10-p\) cards to put a "-" sign before. The sum of the numbers on the board is updated by adding the sum of the numbers on the \(p\) cards and subtracting the sum of the numbers on the remaining \(10-p\) cards.

2. **Formulating the Problem:**
   Let the cards be represented by the set \( \{2^0, 2^1, 2^2, \ldots, 2^9\} \). The goal is to find the greatest absolute value of the number on the board after several rounds, regardless of Sasha's moves.

3. **Analyzing the Sum:**
   The sum of all the cards is:
   \[
   S = 2^0 + 2^1 + 2^2 + \ldots + 2^9 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512
   \]
   This is a geometric series with the first term \(a = 1\) and common ratio \(r = 2\). The sum of the first \(n\) terms of a geometric series is given by:
   \[
   S = a \frac{r^n - 1}{r - 1}
   \]
   Substituting \(a = 1\), \(r = 2\), and \(n = 10\):
   \[
   S = \frac{2^{10} - 1}{2 - 1} = 2^{10} - 1 = 1023
   \]

4. **Calculating the Maximum Absolute Value:**
   In each round, Sasha can choose \(p\) cards to be positive and \(10-p\) cards to be negative. The sum on the board after one round can be expressed as:
   \[
   \text{New sum} = \sum_{i \in P} 2^i - \sum_{j \in N} 2^j
   \]
   where \(P\) is the set of indices of the \(p\) cards chosen to be positive, and \(N\) is the set of indices of the \(10-p\) cards chosen to be negative.

   The maximum possible value of the sum on the board after one round is achieved when all cards are positive:
   \[
   \text{Max sum} = 1023
   \]
   The minimum possible value of the sum on the board after one round is achieved when all cards are negative:
   \[
   \text{Min sum} = -1023
   \]

5. **Greatest Absolute Value:**
   The greatest absolute value of the number on the board after several rounds, regardless of Sasha's moves, is the maximum of the absolute values of the possible sums:
   \[
   \boxed{1023}
   \]