To visit a place or person repeatedly and frequently; to frequent (often with a nuance of persistence or insistence).
学生のころ、彼はあの喫茶店に通いつめて勉強した。
When he was a student, he used to frequent that café to study.
ファンが有名人の自宅に通いつめるのは問題だ。
Fans repeatedly showing up at a celebrity's home is a problem.
彼女は何年もその古本屋に通いつめて希少本を集めている。
She has frequented that secondhand bookstore for years, collecting rare books.