What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2, -10, 5)$ and the other on the sphere of radius 87 with center $(12, 8, -16)$?
1. **Identify the centers and radii of the spheres:**
   - Sphere 1: Center at $(-2, -10, 5)$ with radius $19$.
   - Sphere 2: Center at $(12, 8, -16)$ with radius $87$.

2. **Calculate the distance between the centers of the two spheres using the distance formula:**
   \[
   d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
   \]
   Substituting the given coordinates:
   \[
   d = \sqrt{(12 - (-2))^2 + (8 - (-10))^2 + (-16 - 5)^2}
   \]
   Simplify the expressions inside the square root:
   \[
   d = \sqrt{(12 + 2)^2 + (8 + 10)^2 + (-16 - 5)^2}
   \]
   \[
   d = \sqrt{14^2 + 18^2 + (-21)^2}
   \]
   Calculate the squares:
   \[
   d = \sqrt{196 + 324 + 441}
   \]
   Sum the values inside the square root:
   \[
   d = \sqrt{961}
   \]
   Simplify the square root:
   \[
   d = 31
   \]

3. **Determine the largest possible distance between a point on each sphere:**
   - The farthest distance occurs when the points are on the line connecting the centers and are on the outermost points of each sphere.
   - This distance is the sum of the distance between the centers and the radii of both spheres:
   \[
   \text{Maximum distance} = d + r_1 + r_2
   \]
   Substituting the values:
   \[
   \text{Maximum distance} = 31 + 19 + 87
   \]
   Calculate the sum:
   \[
   \text{Maximum distance} = 137
   \]

The final answer is $\boxed{137}$