How many mipmaps does a texture have in OpenGL

Nevermind that I'm the one who created the texture in the first place and I should know perfectly well how many mipmaps I loaded/generated for it. I'm doing this for a unit test. There doesn't seem to be a glGetTexParameter parameter to find this out. The closest I've come is something like this:

int max_level;
glGetTexParameter( GL_TEXTURE_2D, GL_TEXTURE_MAX_LEVEL, &max_level );
int max_mipmap = -1;
for ( int i = 0; i < max_level; ++i )
{
    int width;
    glGetTexLevelParameter( GL_TEXTURE_2D, i, GL_TEXTURE_WIDTH, &width );
    if ( 0 == width )
    {
        max_mipmap = i-1;
        break;
    }
)

Anyhow, glGetTexLevelParameter() will return 0 width for a nonexistent mipmap if I'm using an NVidia GPU, but with Mesa, it returns GL_INVALID_VALUE, which leads me to believe that this is very much the Wrong Thing To Do.

How do I find out which mipmap levels I've populated a texture with?