In spite of its "recreational" aspect, this question appears to me to be research-level and (I hope) clearly formulated and tagged.

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**Edit 4/4/20:** You can find a related question with the same motivator as [**posed**](https://twitter.com/gregeganSF/status/1245603565978017794) by Greg Egan:

[![enter image description here](https://i.stack.imgur.com/QtnAe.png)](https://i.stack.imgur.com/QtnAe.png)

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**Background:**

Here is a relatively straightforward question from a recent [**538 Riddler**](https://fivethirtyeight.com/features/can-you-tell-when-the-snow-started/):

> 
> Last week, you started with a fair 6-sided die and rolled it six times, recording the results of each roll. You then wrote these numbers on the six faces of *another*, unlabeled fair die. For example, if your six rolls had been 3, 5, 3, 6, 1 and 2, then your second die wouldn’t have had a 4 on it; instead, it would have two 3s.
> 
> 
> Next, you rolled this second die six times. You took those six numbers and wrote them on the faces of yet *another* fair die, and you continued this process of generating a new die from the previous one.
> 
> 
> Eventually, you’d have a die with the same number on all six faces. What was the average number of rolls it would take to reach this state?
> 
> 
> 

The link above contains an exact solution, but there is also a follow-up with some work done, yet which lacks a closed form solution.

**My Question:**

> 
> What is the precise answer for the version of this question in which you start with a fair $N$-sided die?
> 
> 
> 

The 538 piece above includes a [**link**](https://colab.research.google.com/drive/1VCAvgIP2kCxWx7UIsN31n08x0Ue_hIrv) to work from Angela Zhou, who also [**tweets**](https://twitter.com/dancingelv/status/1244642903688953857):
[![enter image description here](https://i.stack.imgur.com/Ib04k.png)](https://i.stack.imgur.com/Ib04k.png)