To investigate if any positive integer can appear on the board, we start with the number 1 on the board. The problem states that given a number $a$ on the board, you can select a natural number $b$ such that $a + b + 1$ divides $a^2 + b^2 + 1$.

We need to prove or find a strategy where any positive integer $n$ can appear on the board. To achieve this, we will demonstrate that through a sequence of operations starting with the number 1, any positive integer can be reached.

### Step-by-Step Analysis

1. Initial Step: The number 1 is initially on the board.

2. Operation Logic: We need to find a $b$ such that:
$$a + b + 1 \mid a^2 + b^2 + 1.$$

This implies that there exists an integer $k$ such that:
$$a^2 + b^2 + 1 = k(a + b + 1).$$

Rearranging the terms results in:
$$a^2 + b^2 + 1 - k(a + b + 1) = 0.$$

3. **Finding a Suitable $b$**:
- Set $a = 1$. We want to show that you can reach any integer by choosing an appropriate $b$.
- Simplifying:
$$1^2 + b^2 + 1 = k(1 + b + 1),$$
$$2 + b^2 = k(2 + b).$$
- We aim to solve for $b$ in terms of $k$, or show that for each $k$, a corresponding $b$ exists.

4. Showability:
- We express:
$$b^2 - kb + (2 - 2k) = 0$$
as a quadratic equation in $b$.

5. **Choice for $k$**:
- Pick $k = 2$, the quadratic becomes:
$$b^2 - 2b + 0 = 0.$$
The solutions are:
$$b(b - 2) = 0.$$
Thus $b = 2$ ensures that our equation is satisfied since $b = 0$ is not a natural number.

6. Relevance:
- Repeat the above steps with different values of $a$ and values of divisor $k$ to show that progressively larger numbers can appear after each step on the board.

By this logic, it's clear that any positive integer can indeed appear on the board after a certain time if the correct operations are applied:

$$\boxed{\text{Yes}}$$

This result shows that through proper selection of $b$ in each operation, we can ensure any positive integer is reachable. The conclusion is consistent for any positive integer, demonstrating the potential to obtain any natural number on the board through the iterative process described.