There are no league teams still in existence that has never been...
relegated. Arsenal is the team to have gone the longest without relegation.Arsenal were relegated in 1912/1913 from Division One as was along with Notts County. Football was suspended officially until the war was over in which time they changed to Arsenal f.c. They were voted back into the top division in 1919 and have remained there ever since, longer than anyone else and since then have never been relegated from top flight football.
One could say that Everton have never been relegated, but it just isn't entirely true. They were relegated to the second division in 1931 and came straight back up the following year. They were also relegated in 1951, and came back to the top flight in 1954.
They have spent more seasons in English Footballs top division than any other club and have the second longest unbroken run in the top flight besides Arsenal FC.