I read here that "for the unit hypersphere, the hyper-surface area reaches a maximum and then decreases towards 0 as n increases...As a result, the seven-dimensional unit hypersphere has maximum hyper-surface area (Le Lionnais 1983; Wells 1986, p. 60)."
How closely do the hyper-surface areas of unit spheres of dimension > 7 approach 4π?
From the chart I saw, it looks like dimension 13 may be the closest.