To find the value of \(1 + \sec^2 30^\circ\) without using tables, follow these steps:

1. Recall the trigonometric identity:
\[
\sec^2 \theta = 1 + \tan^2 \theta
\]

2. Use this identity for \(\theta = 30^\circ\):
\[
\sec^2 30^\circ = 1 + \tan^2 30^\circ
\]

3. Find \(\tan 30^\circ\):
\[
\tan 30^\circ = \frac{1}{\sqrt{3}}
\]

4. Square \(\tan 30^\circ\):
\[
\tan^2 30^\circ = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3}
\]

5. Substitute into the identity:
\[
\sec^2 30^\circ = 1 + \frac{1}{3} = \frac{4}{3}
\]

6. Calculate \(1 + \sec^2 30^\circ\):
\[
1 + \sec^2 30^\circ = 1 + \frac{4}{3} = \frac{3}{3} + \frac{4}{3} = \frac{7}{3}
\]

7. Convert to mixed fraction (if needed):
\[
\frac{7}{3} = 2 \frac{1}{3}
\]

Therefore, the correct answer is:

A. \(2 \frac{1}{3}\)