Let $N$ be the positive integer with 1998 decimal digits, all of them 1; that is,
\[N=1111\cdots 11.\]
Find the thousandth digit after the decimal point of $\sqrt N$.
To find the thousandth digit after the decimal point of \(\sqrt{N}\), where \(N\) is a number with 1998 digits, all of them being 1, we can proceed as follows:

1. **Express \(N\) in a more manageable form:**
   \[
   N = 111\ldots111 \quad \text{(1998 digits)}
   \]
   This can be written as:
   \[
   N = \frac{10^{1998} - 1}{9}
   \]

2. **Approximate \(\sqrt{N}\):**
   We need to find \(\sqrt{N}\). Using the expression for \(N\):
   \[
   \sqrt{N} = \sqrt{\frac{10^{1998} - 1}{9}}
   \]
   For large \(n\), \(\sqrt{10^{2n} - 1} \approx 10^n\). Therefore:
   \[
   \sqrt{N} \approx \sqrt{\frac{10^{1998}}{9}} = \frac{10^{999}}{\sqrt{9}} = \frac{10^{999}}{3}
   \]

3. **Decimal expansion of \(\frac{10^{999}}{3}\):**
   To find the thousandth digit after the decimal point of \(\frac{10^{999}}{3}\), we need to understand the decimal expansion of \(\frac{1}{3}\):
   \[
   \frac{1}{3} = 0.\overline{3}
   \]
   Therefore:
   \[
   \frac{10^{999}}{3} = 10^{999-1} \times \frac{10}{3} = 10^{998} \times 3.\overline{3} = 3.\overline{3} \times 10^{998}
   \]

4. **Determine the thousandth digit:**
   The decimal expansion of \(3.\overline{3} \times 10^{998}\) will be:
   \[
   3.\underbrace{333\ldots333}_{998 \text{ threes}}333\ldots
   \]
   The thousandth digit after the decimal point is the 1000th digit in the sequence of threes.

Therefore, the thousandth digit after the decimal point of \(\sqrt{N}\) is \(3\).

The final answer is \(\boxed{3}\)