I know there are a few questions about this sort of thing already, so excuse me if this is a duplicate.
I noticed something interesting when playing around with primorials and related quantities. I got the idea to do so after seeing the popular proof by contradiction that there are infinitely many primes. Below is a not-really-ordered list of primes that I found while messing around. First off, we have a little pattern involving primorial+1, but it stops at $13\text{#}=2\cdot3\cdot5\cdots13$ $$\begin{align} 3&=2+1=2!+1\\ 7&=2\cdot3+1=3!+1\\ 31&=2\cdot3\cdot5+1\\ 211&=2\cdot3\cdot5\cdot7+1\\ 2311&=2\cdot3\cdot5\cdot7\cdot11+1\\ \color{red}{59\cdot509}&\color{red}{\,\,=13\text{#}+1} \end{align}$$ Then, since the primorial method seemed to fail, I began excluding factors from the primorials, and multiplying by primes that I had previously found: $$\begin{align} 929&=2\cdot3\cdot5\cdot31\color{blue}{-1}\\ 1303&=2\cdot3\cdot7\cdot31+1\\ 1373611&=2\cdot3\cdot5\cdot7\cdot31\cdot211+1\\ 15109709&=2\cdot3\cdot5\cdot7\cdot11\cdot31\cdot211\color{blue}{-1}\\ 41163989&=2\cdot3\cdot5\cdot7\cdot211\cdot929\color{blue}{-1}\\ 255216737&=2\cdot3\cdot7\cdot31\cdot211\cdot929\color{blue}{-1}. \end{align}$$ I then noticed that the following were also primes: $$\begin{align} 23&=4!-1\\ 719&=6!-1\\ 5039&=7!-1, \end{align}$$ As well as $$\begin{align} 967&=2\cdot3\cdot7\cdot23+1\\ 30197&=2\cdot3\cdot7\cdot719\color{blue}{-1}\\ 211639&=2\cdot3\cdot7\cdot5039+1. \end{align}$$ I found these all in about 30 minutes, and I'm sure I could find a lot more if I tried.
I am a firm believer that when it comes to number theory, almost nothing is a coincidence. Intuitively, the occurrences of these primes makes sense: when you add or subtract $1$ to a highly composite number $N$, the result is a lot less likely to have just as many divisors as $N$, and a lot of the time, at least one of $N\pm1$ ends up being prime (Mersenne primes are a great example). However, I lack any deeper understanding of this.
Are there any theorems which have to do with constructing primes from highly composite numbers? Given any finite set of primes, is their product more likely to be $1$ away from a prime than any given integer with the same number of digits?
Really, this question is a resource request on the construction and occurrence of primes.
And of course, if you find any more nice examples using my 'method' above, feel free to add them as an answer.