Thirty identical toothpicks were used to create the figure below. There are over $25$ triangles in the figure. What is the fewest number of toothpicks that could be removed so that no triangles remain?

To find the fewest number of toothpicks that could be removed so that no triangles remain, let's start by analyzing the figure.

Looking at the figure, we notice that each toothpick is an edge of either one or two triangles. Since there are over 25 triangles, there must be at least $\frac{25}{2} = 12.5$ toothpicks that are an edge of two triangles (since each triangle has three edges, and each toothpick is counted twice when it is shared by two triangles).

Since the number of toothpicks is a whole number, there must be at least 13 toothpicks that form the edges of two triangles.

Now, consider removing one toothpick that forms the edge of two triangles. By doing so, we remove two triangles from the figure. However, removing this toothpick also removes one of the edges from each of the adjacent triangles, making them incomplete and removing those triangles as well. So removing just one toothpick that forms the edge of two triangles, actually removes four triangles total.

Therefore, by removing 13 toothpicks, we would remove $13 \times 4 = 52$ triangles, which is more than the number of triangles in the figure.

This means that a minimum of 14 toothpicks must be removed to eliminate all the triangles.

But does removing 14 toothpicks eliminate all triangles?

To answer this question, let's create a smaller figure using just 5 toothpicks to see if it's possible to remove all triangles. It's clear that no matter which toothpick you remove, at least one triangle will always remain. Therefore, it is not possible to remove all triangles from the figure using only 14 toothpicks.

Hence, the fewest number of toothpicks that could be removed so that no triangles remain is 15.