Let $L$ be the length of the altitude to the hypotenuse of a right triangle with legs $5$ and $12$.  Find the least integer greater than $L$.
1. **Calculate the area of the right triangle:**
   The legs of the right triangle are given as \(5\) and \(12\). The area \(A\) of a right triangle can be calculated using the formula:
   \[
   A = \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2
   \]
   Substituting the given values:
   \[
   A = \frac{1}{2} \times 5 \times 12 = 30
   \]

2. **Determine the length of the hypotenuse:**
   Using the Pythagorean theorem, the hypotenuse \(c\) of a right triangle with legs \(a\) and \(b\) is given by:
   \[
   c = \sqrt{a^2 + b^2}
   \]
   Substituting the given values:
   \[
   c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13
   \]

3. **Relate the area to the altitude:**
   The area of the triangle can also be expressed in terms of the hypotenuse \(c\) and the altitude \(h\) to the hypotenuse:
   \[
   A = \frac{1}{2} \times c \times h
   \]
   Substituting the known values:
   \[
   30 = \frac{1}{2} \times 13 \times h
   \]

4. **Solve for the altitude \(h\):**
   Rearrange the equation to solve for \(h\):
   \[
   30 = \frac{13h}{2} \implies 60 = 13h \implies h = \frac{60}{13}
   \]

5. **Find the least integer greater than \(h\):**
   Calculate the value of \(h\):
   \[
   h = \frac{60}{13} \approx 4.615
   \]
   The least integer greater than \(4.615\) is \(5\).

The final answer is \(\boxed{5}\).