Below I list some websites that I visit very often. They are not very neatly categorized. I may do so at some point in time.
Cardiff University
University of Vienna
Tezpur University
The Abdus Salam International Centre for Theoretical Physics
Chennai Mathematical Institute
Harish-Chandra Research Institute
The Institute of Mathematical Sciences
Gonit Sora
Arxiv.CO
Arxiv.NT
Arxiv.GM
Ron Graham Archives
George E. Andrews
Rupam Barman
Nayandeep Deka Baruah
Bruce C. Berndt
Mihai Ciucu
Ilse Fischer
Christian Krattenthaler
M. Ram Murty
Anupam Saikia
Fernando Rodriguez Villegas
Pankaj Jyoti Mahanta
Khimpi Dutta
Terence Tao
Tim Gowers
Great Bong
Onionesque Reality
Tanya Khovanova
The Aperiodical
Math with Bad Drawings
The value of the variable: H. A. Helfgott
Luit-Pariya
Gil Kalai
Peter Cameron
Igor Pak
British Combinatorial Committee
Kaneenika Sinha
Academia Obscura
Pankaj Jyoti Mahanta
Viviane Pons
Mukunth V
Integers
Bulletin of the Australian Mathematical Society
The Electronic Journal of Combinatorics
Journal of Integer Sequences
Journal of Number Theory
The Ramanujan Journal
The Mathematical Intelligencer
Annals of Combinatorics
Discrete Mathematics
European Journal of Combinatorics
The Journal of Irreproducible Results
Annals of Improbable Research
Proceedings of the National Institute of Science
Journal of Universal Rejection