Let the centers of your 3 spheres of radius 1 be C1,C2,C3. Let the center of the sphere of radius 2 be C4. These 4 points forms a tetrahedron with equilateral triangle base of length 2 ( C1C2=C2C3=C3C1=2) , and a side length of 3 (C4C1=C4C2=C4C3=3). Let O be the center of the equilateral base, then OC1=2/3√3
Hence, the distance from the plane to the top is 3+√69/3.