V OBM

We can place $6$ spheres with their centers coplanar with the fixed sphere. Then we can place $3$ more above and $3$ more below as shown above. Thus $12$ can be achieved.



Take $O$ to be the center of the fixed sphere. Another sphere touching it blocks off a conical solid angle as shown. The angle between the center line of the cone and the surface is $30^{\circ }$. It's not hard to show that for an angle $\theta$ the solid angle is $2\pi (1-\cos \theta )$, so for $\theta =30^{\circ }$, it is $\pi (2-\sqrt{3})$. Thus we can have at most $\frac{4\pi }{\pi (2-\sqrt{3})}=4(2+\sqrt{3})<15$ spheres touching the fixed sphere. Hence at most $14$.