Let $x_i\  (i = 1, 2, \cdots 22)$ be reals such that $x_i \in [2^{i-1},2^i]$. Find the maximum possible value of

$$(x_1+x_2+\cdots +x_{22})(\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_{22}})$$
1. **Identify the range of each \( x_i \):**
   Given \( x_i \in [2^{i-1}, 2^i] \) for \( i = 1, 2, \ldots, 22 \).

2. **Expression to maximize:**
   We need to maximize the expression:
   \[
   (x_1 + x_2 + \cdots + x_{22}) \left( \frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_{22}} \right)
   \]

3. **Convexity and boundary values:**
   By compactness and continuity, a unique maximum exists. The expression is convex in each variable, hence the maximum occurs when each \( x_i \) is either \( 2^{i-1} \) or \( 2^i \).

4. **Case analysis:**
   - **Case 1:** All \( x_i = 2^{i-1} \)
     \[
     \sum_{i=1}^{22} x_i = \sum_{i=1}^{22} 2^{i-1} = 2^0 + 2^1 + \cdots + 2^{21} = 2^{22} - 1
     \]
     \[
     \sum_{i=1}^{22} \frac{1}{x_i} = \sum_{i=1}^{22} \frac{1}{2^{i-1}} = 2^0 + 2^{-1} + \cdots + 2^{-21} = 2 - 2^{-21}
     \]
     \[
     \text{Expression} = (2^{22} - 1)(2 - 2^{-21}) \approx 2^{23}
     \]

   - **Case 2:** All \( x_i = 2^i \)
     \[
     \sum_{i=1}^{22} x_i = \sum_{i=1}^{22} 2^i = 2^1 + 2^2 + \cdots + 2^{22} = 2^{23} - 2
     \]
     \[
     \sum_{i=1}^{22} \frac{1}{x_i} = \sum_{i=1}^{22} \frac{1}{2^i} = 2^{-1} + 2^{-2} + \cdots + 2^{-22} = 1 - 2^{-22}
     \]
     \[
     \text{Expression} = (2^{23} - 2)(1 - 2^{-22}) \approx 2^{23}
     \]

   - **Case 3:** There exists some \( 1 \leq k \leq 21 \) such that for \( 1 \leq i \leq k \), \( x_i = 2^{i-1} \), and for \( k+1 \leq i \leq 22 \), \( x_i = 2^i \)
     \[
     \sum_{i=1}^{k} x_i = \sum_{i=1}^{k} 2^{i-1} = 2^k - 1
     \]
     \[
     \sum_{i=k+1}^{22} x_i = \sum_{i=k+1}^{22} 2^i = 2^{k+1} + 2^{k+2} + \cdots + 2^{22} = 2^{23} - 2^{k+1}
     \]
     \[
     \sum_{i=1}^{22} x_i = (2^k - 1) + (2^{23} - 2^{k+1}) = 2^{23} - 2^k - 1
     \]
     \[
     \sum_{i=1}^{k} \frac{1}{x_i} = \sum_{i=1}^{k} \frac{1}{2^{i-1}} = 2 - 2^{-k}
     \]
     \[
     \sum_{i=k+1}^{22} \frac{1}{x_i} = \sum_{i=k+1}^{22} \frac{1}{2^i} = 2^{-k-1} + 2^{-k-2} + \cdots + 2^{-22} = 2^{-k}(1 - 2^{-21}) = 2^{-k} - 2^{-22}
     \]
     \[
     \sum_{i=1}^{22} \frac{1}{x_i} = (2 - 2^{-k}) + (2^{-k} - 2^{-22}) = 2 - 2^{-22}
     \]
     \[
     \text{Expression} = (2^{23} - 2^k - 1)(2 - 2^{-22})
     \]

5. **Maximizing the expression:**
   To maximize the expression, we need to minimize \( 2^k + 2^{22-k} \). This occurs when \( k = 11 \):
   \[
   2^k + 2^{22-k} = 2^{11} + 2^{11} = 2^{12}
   \]
   \[
   \text{Expression} = \frac{(2^{23} - 2^{11} - 1)^2}{2^{22}} \approx 2^{24}
   \]

The final answer is \( \boxed{2^{24}} \).