To find the maximum value of $m$, we need to evaluate the constraints given by the problem, specifically that none of the subsets $A_1, A_2, \ldots, A_m$ is contained in any other subset. Each subset $A_i$ has exactly $i$ elements.

Let's consider the key points to derive the maximum value of $m$:

1. Condition on Subsets:
- There are $n$ distinct elements in total.
- Subset $A_1$ contains exactly 1 element, $A_2$ contains exactly 2, and so on, up to $A_m$, which contains exactly $m$ elements.
- No subset $A_i$ is contained within another subset $A_j$.

2. Binomial Coefficient Representation:
- The total number of ways to choose subsets of varying sizes from a set with $n$ elements is given by the binomial coefficients.
- Therefore, for subset $A_i$, a key constraint is that $\binom{n}{i}$ must account for at least $i$ elements being distinct in the choice so that none of the subsets are contained within another.

3. **Deriving the Maximum $m$:**
- To adhere to the condition that none of these subsets is contained in any other subset, the sum of sizes of all subsets cannot exceed $n$, the total number of distinct elements:
$$1 + 2 + \dots + m \leq n$$
- The left hand side is the sum of the first $m$ positive integers, which is $\frac{m(m + 1)}{2}$.
- So, we form the inequality:
$$\frac{m(m + 1)}{2} \leq n$$

4. Solving the Inequality:
- Multiply both sides by 2 to clear the fraction:
$$m(m + 1) \leq 2n$$
- This quadratic inequality can be solved for $m$ by rearranging and setting up the quadratic equation:
$$m^2 + m - 2n = 0$$
- Using the quadratic formula:
$$m = \frac{-1 \pm \sqrt{1 + 8n}}{2}$$
- Since $m$ has to be a positive integer and we want the maximum $m$, we use the floor function:
$$m = \left\lfloor \frac{-1 + \sqrt{1 + 8n}}{2} \right\rfloor$$

Thus, the maximum value of $m$ is:
$$\boxed{\left\lfloor \frac{-1 + \sqrt{1 + 8n}}{2} \right\rfloor}$$