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 $14$.