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# [BQN], 23 bytes

    {𝕩{𝕊⍟(𝕗>·+´∘÷1+↕)𝕩+1}1}

This recursion is more complicated than it should, but I managed to keep it purely functional. I use a modifier to mimic passing a left argument, that's why we have the 𝕗 inside the recursion test.

# [BQN], 17 bytes

    ⌊0.5+·⋆⊢-0.57722˙

I don't know if it is fair to use an approximation, but the harmonic series' partial sums are given by the Euler-Maclaurin formula, so we can do:

$H_k \approx \ln k + \gamma > n$

So we solve for k and take the nearest integer. I have tuned the Euler–Mascheroni constant value so I can reproduce the recursion (although I hit CBQN's recursion limit for n=9 already):

[BQN online REPL][Try]

[BQN]: https://mlochbaum.github.io/BQN/
[Try]: https://mlochbaum.github.io/BQN/try.html#code=QSDihpAg4oyKMC41K8K34ouG4oqiLTAuNTc3MjLLmQpSIOKGkCB78J2VqXvwnZWK4o2fKPCdlZc+wrcrwrTiiJjDtzEr4oaVKfCdlakrMX0xfQoKKEHCqOKLiMK34oiowrTCtyhB4omhUinCqMKvM+KGk+KKoikgMSvihpUxMAo=