The number of n-qubit stabilizer states is:
$$|\text{StabilizerStates(n)}| = 2^n \prod_{k=0}^{n-1} (2^{n-k}+1)$$
This is a bit cumbersome, so it's nicer to use this approximation that you get by dropping the +1 in each term of the product:
$$|\text{StabilizerStates(n)}| \approx \sqrt{2}^{(n+1)(n+2)}$$
The true value is always within $\pm50\%$ of this value. In fact, as $n$ increases, the ratio between the approximation and the exact value seems to converge to a constant around 1.19211. Is there a simple expression for this constant in terms of other constants? Similar to the $\sqrt{2 \pi}$ in Stirling's approximation.
n approx_ratio exact
0 0.5 1
1 0.75 6
2 0.9375 60
3 1.0546875 1080
4 1.12060546875 36720
5 1.1556243896484375 2423520
6 1.1736810207366943 315057600
7 1.1828504037111998 81284860800
8 1.1874709131006966 41780418451200
9 1.1897901922278464 42866709330931200
10 1.1909520967124438 87876754128408960000
11 1.191533616290917 360118938418219918080000
12 1.1918245180527065 2950814581398894008747520000
13 1.1919700044440704 48352047730802277227336862720000
14 1.192042756519537 1584496604138390624739828991334400000
15 1.1920791347774875 103844738442021844764198912434073907200000
16 1.1920973244615507 13611345246549571280622608248383803312332800000
17 1.1921064194423587 3568159711001983912930094261880820503114794188800000
18 1.1921109669674574 1870750454881230145710119120561495381578055445245952000000
19 1.1921132407386803 1961627770478450543728425287200131704224354222665109856256000000
20 1.1921143776264602 4113835525369964471582242004752905004040921355479025795486695424000000
21 1.1921149460708924 17254685027072394202944227133987137975878860686813810768164619537055744000000
22 1.1921152302932438 144742823364949756734759973178435845495837169546013201147933409094741184675840000000
23 1.1921153724044535 2428381901529255627786036247688771078897978285976424506536729053865655508884827013120000000
24 1.192115443460067 81482980241657907031675119394680521707991005130229061660569243613817582639612341613152174080000000
25 1.1921154789878756 5468230402318068100829143342991616244111659492489854648284878853401039901805670925263253087943393280000000
Amazing is $$\frac 14 \left(-1;\frac{1}{2}\right)_{\infty }=5 \left(\cos \left(\frac{10 \pi }{51}\right)-\sin \left(\frac{10 \pi }{51}\right)\right)+1.10\times 10^{-7}$$ $$\frac 14 \left(-1;\frac{1}{2}\right)_{\infty }=2^{\frac{2+\sqrt{46}}{20 \sqrt{3}}}+7.39\times 10^{-9}$$ which $\cdots\cdots$ does not mean anything.