There are seven cards in a hat, and on the card $k$ there is a number $2^{k-1}$, $k=1,2,...,7$. Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds $124$. What is the most probable sum he can get?
1. **Identify the numbers on the cards:**
   The numbers on the cards are given by \(2^{k-1}\) for \(k = 1, 2, \ldots, 7\). Therefore, the numbers on the cards are:
   \[
   2^0, 2^1, 2^2, 2^3, 2^4, 2^5, 2^6 = 1, 2, 4, 8, 16, 32, 64
   \]

2. **Determine the sum of all cards:**
   The sum of all the numbers on the cards is:
   \[
   1 + 2 + 4 + 8 + 16 + 32 + 64 = 127
   \]

3. **Identify the stopping condition:**
   Solarin stops picking cards when the sum of the numbers on the cards in his hand exceeds 124. Therefore, the possible sums he can get are 125, 126, and 127.

4. **Analyze the possible sums:**
   - **Sum = 125:**
     To get a sum of 125, the sum of the numbers on the cards picked must be 125. This means the last card picked must be 2 (since 127 - 2 = 125).
   - **Sum = 126:**
     To get a sum of 126, the sum of the numbers on the cards picked must be 126. This means the last card picked must be 1 (since 127 - 1 = 126).
   - **Sum = 127:**
     To get a sum of 127, the sum of the numbers on the cards picked must be 127. This means the last card picked can be any of the remaining cards (3, 4, 5, 6, 7).

5. **Calculate the probabilities:**
   - **Probability of stopping at 125:**
     The probability of stopping at 125 is the probability of picking the card with the number 2 as the last card. There are 7 cards, so the probability is:
     \[
     \frac{1}{7}
     \]
   - **Probability of stopping at 126:**
     The probability of stopping at 126 is the probability of picking the card with the number 1 as the last card. There are 7 cards, so the probability is:
     \[
     \frac{1}{7}
     \]
   - **Probability of stopping at 127:**
     The probability of stopping at 127 is the probability of picking any of the remaining cards (3, 4, 5, 6, 7) as the last card. There are 5 such cards, so the probability is:
     \[
     \frac{5}{7}
     \]

6. **Determine the most probable sum:**
   Since the probability of stopping at 127 is the highest (5/7), the most probable sum Solarin can get is 127.

The final answer is \(\boxed{127}\).