Given a sequence defined as $x_1 = 1$, and for $k \geq 1$:
$$x_{k+1} = \begin{cases} x_k + d & \text{if } a \text{ does not divide } x_k \\ \frac{x_k}{a} & \text{if } a \text{ divides } x_k \end{cases}$$
we need to determine the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

### Analysis

1. Initial Observations:
- The sequence starts at $x_1 = 1$.
- We apply the operation $x_k + d$ as long as $x_k$ is not divisible by $a$.

2. Divisibility Rule:
- Whenever $x_k$ becomes divisible by $a$, we divide it by $a$.
- We aim to explore how deeply $x_k$ can be divisible by $a$, or how large $n$ can be such that $a^n \mid x_k$.

3. Operation Analysis:
- Each time $a \mid x_k$, we reduce the power of $a$ in $x_k$ by one (i.e., $x_k \to x_k/a$).
- This reduction can occur only if, between consecutive $a \mid x_k$ conditions, the additions $x_{k} + d$ consistently reach a point $x_k \equiv 0 \pmod{a}$.

4. Balancing Act:
- We require that adding $d$, which is coprime to $a$, should eventually lead back to a number divisible by higher powers of $a$.

5. Rational Argument:
- If $a^n \mid x_k$ for some $n$, then undergoing the reduction $x_k/a$ for reaching $a^n$ implies:
- Possible continuous multiplication of $a$ $(n$ times) without returning to situation without $a \mid x_k$.
- The key reaches through exploration that achieving $x_k$ reduces by dividing $a$ into $d^1, d^2, \ldots$, up to $d^n$.

6. Critical Insight:
- Since $a^n \times x_1 = a^n \times 1 = a^n$, and our $x_k$ grows through increments of $d$,
- The critical component driving when $x_k \equiv 0 \pmod{a^n}$ is fundamentally bound by how additions of $d$ can fill these slots.
- We resolve that the greatest $n$ for which this manipulation of $x_k$ evolves is encapsulated by:
$$n = \lceil \log_a d \rceil$$

Hence, the greatest integer $n$ such that there exists some $x_k \equiv 0 \pmod{a^n}$ is:
$$\boxed{\lceil \log_a d \rceil}$$