According to Wikipedia, the infinity that are the Rolling Stones had singles reach numbers 1,2,3,5,6,7,8,9,10,11,12,13,14, and 15 on the UK charts. In the US, it’s similar, 1,2,3,5,6,7,8,9 and 10. No number 4’s in either case.
Can any other group beat this?
The best I can do is the Herman’s Hermits who hit numbers 1,2,3,4,5,6,7,8,9,10 and 11 on the New Zealand charts.
You’re only looking for different top rankings, right (not the total number of 1s, for example)?
The obvious answer, The Beatles, have: 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 17, 19. That includes 20 different #1s. They only miss a #9 hit from the Top 10.
Data from here: U2 | Biography, Music & News | Billboard
And we plunge immediately down the rabbit hole of what counts.
If you look at the Wikipedia you see that many of those songs were b-sides that got counted by some arcane measure of airplay. Otherwise they have lots of number ones rather than a distribution. That’s why I didn’t include them.
But what the heck. Give them nine of ten as well.
And thanks for that Billboard link. Makes it almost too easy. 
Madonna is now in the lead with 1,2,3,4,5,6,7,8,9,10 and 11.
U2 hit all 10 spots on the UK charts. In fact they hit 14 of the top 15, only missing a #11. In their native Ireland, they’re missing a #9, despite having 41 Top 10 hits.
Elton John hit all Top 12 spots on the UK charts.
Frank Sinatra had singles that peaked in each of the top 28 positions on the US charts. Problem is his careers spanned multiple chart formats. Is a Billboard chart from 1943 measuring the same thing as the charts from 1970? The Hot 100 didn’t start until 1958, Sinatra’s solo career started in the early 1940s. So he may or may not count, for the purposes of this thread.
Queen in Ireland 1,2,3,4,5,6,7,8,9,10,11,13,14,15
Madonna has 1 through 11 and 14.
According to here.