I only recently found out that it is an open question whether the Euler-Mascheroni constant is irrational or not and whether is transcendental or not. I found that quite surprising and looked for lecture notes on the topic. I found these lectures by Michel Waldschmidt with the title: “An introduction to irrationality and transcendence methods.”:

• https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/AWSLecture1.pdf
• https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/AWSLecture2.pdf
• https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/AWSLecture3.pdf
• https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/AWSLecture5.pdf

There is also a series of video lectures by the same author on Youtube:

The title image of this post is by Arthur Baelde on Wikimedia Commons. See https://commons.wikimedia.org/wiki/File:First_irrational_numbers.svg