A certain class of books is defined as follows: (1) the book was kept for years in a cafe or mathematics library; (2) the primary contents are research problems and comments, handwritten by resident and visiting mathematicians; (3) the book still exists. Examples include the Lwów and Wrocław Scottish Books in Poland and the Boneyard Book in the University of Illinois Archives. What are some others?
The library of the math institute in Göttingen contains many manuscripts that fulfill more or less your requirements. Most prominently are the 29 volumes of the Felix Klein Protokolle. They are concerned with research problems mostly in the early years; in the later years they contain more undergraduate topics.
It now appears that the Boneyard Book in the University of Illinois Archives may be the only American book of its kind. Attempts to locate a Longhorn Problem Book in Texas have been unsuccessful, and no trace of other books of handwritten problems and comments by resident and visiting mathematicians was found after consulting with more than 20 large American university mathematics librarians.
Regarding the "Ann Arbor" book, mentioned in this month's Notices of AMS (pages 244-247), a librarian at the Bentley Historical Library, University of Michigan, found the Mathematics Club (U or M) Record Book for 1962-1991, which is not a problem book. However, on page 3, there is a note that "The book containing all previous recods of the Math Club was in an automobile driven by J. Cohn. This auto was stolen in Chicago and later turned up burned out and minus the records." Just below, in a different hand, this appears: "Note: the above is incorrect. Vol.1 of the minutes is housed in the Main Library. Vol. II was lost in Chicago and this book is V. 3." In any case, the volume lost in Chicago appears to be the one mentioned by Paul Erdős as the Ann Arbor Problem Book.
A few details about the Boneyard Book may be of interest. An archivist found that Erdős's name appears ten times in the book, along with names of at least thirty other problem posers and commenters. Mysteriously, several pages were missing: 13-14, 29-30, 35-36, 51-56, 59-66.
Mathematics libraries in Göttingen and Utrecht have been mentioned in Answers dated March 4, 2015. Since then, the librarian of the Mathematisches Institut in Göttingen has written that the collections mentioned "are different from the books you are searching for because there you cannot find ongoing handwritten entries referring to research problems associated with comments of different resident or visiting mathematicians." A mathematics collection specialist at the university library in Utrecht writes that there was "a book which was sitting on a table in the mathematics library for about 30 years...is of course highly valued in our department...(however)...In the book mostly students asked questions about subjects or homework they had problems with, and others then wrote answers and explanations. There were of course well-known mathematicians...working in Utrecht, but I do not think visiting mathematicians often wrote in it."
There are indications that there were communal problems books of handwritten research-level problems kept in at least two Canadian university mathematics departments. I hope that someone can tell what happened to them or where they are now kept.