I apologize for the oversight. Let's reanalyze the problem carefully to find the correct infimum of the set of all probabilities \( p \) for which Aaron has a nonzero probability of winning the game.

### Problem Restatement

Aaron and Beren are playing on an infinite complete binary tree:

- Each edge is independently labeled "A" with probability \( p \) and "B" with probability \( 1-p \).
- Aaron starts at the root and wants to follow a path consisting entirely of edges labeled "A".
- Beren wins if the token ever crosses an edge labeled "B".
- Aaron wins if he can avoid ever traversing a "B" edge, which means finding an infinite path labeled only with "A".

### Key Insights

1. **Winning Path and Infinite Paths of "A"**: Aaron wins if there exists an infinite path from the root through the tree with only "A" edges. If Aaron can ensure that there is such a path no matter how Beren plays, then he has a nonzero probability of winning.

2. **Percolation Condition**: If \( p \) is too low, the probability of finding such an infinite path decreases, because the presence of "B"-labeled edges becomes more common, blocking potential paths.

### Critical Observation with Correct Analysis

To correctly solve this problem, we need to consider the expected number of infinite "A"-labeled paths that emanate from the root. 

- If Aaron can ensure that there are enough "A" edges to keep extending paths indefinitely, then he can find a way to win regardless of Beren's moves. This means examining when the expected number of these paths becomes nonzero.

#### Calculating the Probability

1. **Probability of a Path Surviving**: In each step down the tree, Aaron and Beren can choose between two edges. To find a surviving path of "A", Aaron needs to keep avoiding "B" edges. Aaron will always have two choices at each node, as long as at least one "A"-labeled edge exists. 

2. **Probability Calculation for Aaron's Win**: We calculate the critical value \( p \) where the probability of having at least one "A" edge in every branching step is sufficient for an infinite path to survive.

- **Survival Probability Calculation**: The key condition is that Aaron wants each subtree to have a nonzero probability of having all "A" paths:

    - At any node, Aaron chooses a path with an "A" edge. If both children from a node lead to subtrees with nonzero probabilities of having an all "A" path, Aaron can continue.

3. **Recursive Condition**: The probability \( q \) that a subtree has an infinite "A"-path satisfies a recursive relation:

    \[
    q = p(1 - (1 - q)^2)
    \]

    - The equation can be understood as: starting from a node, the probability of reaching a path of only "A" depends on both children having paths of "A". Thus, Aaron succeeds if at least one child has an "A" path.
  
#### Solving for the Critical \( p \)

1. **Critical Probability Threshold**: Solving the recursion relation, \( q = p(1 - (1 - q)^2) \), we need to determine the smallest \( p \) for which \( q > 0 \). This corresponds to a branching process where Aaron can perpetually continue finding paths labeled "A" to infinity.

2. **Calculating**: 

    - Setting \( q = 0 \) provides \( q = p(1 - 1) = 0 \). 

    - Setting \( q = 1 \) provides \( q = p(1 - 0) = p \). 

    The smallest nonzero \( q \) occurs when:
    \[
    1 - (1 - q)^2 = 0 \Rightarrow q = 1 - \sqrt{1-p}
    \]

    For \( q > 0 \), we want \( 1 - \sqrt{1-p} > 0 \), which implies:

    \[
    \sqrt{1-p} < 1 \Rightarrow 1-p > 0 \Rightarrow p > 0
    \]

3. **Conclusion for Threshold \( p \)**:

   - The recursive relation suggests that the critical \( p \) where Aaron can ensure an infinite "A" path in the tree starts at \( p = 1 - \frac{1}{2} = \frac{1}{2} \).

However, this has to be validated against the structure and branching condition. The smallest \( p \) where \( q > 0 \) (using nonzero probability for continuation):

\[
\frac{1}{2}
\]

The rigorous condition suggests Aaron wins for:

\[
\boxed{p > 0}
\]

### Correct Answer

After reevaluating conditions for infinite paths to exist under random binary processes in infinite trees, the infimum of \( p \) ensuring nonzero win probability is:

\[
\boxed{p > 0}
\] 

Thus, any nonzero probability for \( p \) provides Aaron a chance to continue finding infinite "A" paths, establishing a nonzero chance of winning.

Explore the probability theory problem where Aaron aims to find an infinite path labeled with "A" on a binary tree. Learn the critical probability threshold and recursive relation used to determine Aaron's winning chance. Suitable for advanced mathematics enthusiasts and students.

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