The regular maximum for Divine Smite is 5d8, but increases to 6d8 against undead and fiends
As of the 2018 PHB errata, the description of the Divine Smite feature says:
Starting at 2nd level, when you hit a creature with a melee weapon attack, you can expend one spell slot to deal radiant damage to the target, in addition to the weapon’s damage. The extra damage is 2d8 for a 1st-level spell slot, plus 1d8 for each spell level higher than 1st, to a maximum of 5d8. The damage increases by 1d8 if the target is an undead or a fiend, to a maximum of 6d8.
As other answers have pointed out, 5d8 is the regular maximum for the total amount of "extra radiant damage" done by Divine Smite, regardless of the spell slot used. 5d8 is an overall limit for the feature - not just for the additional radiant damage done when using a higher-level spell slot.
The portion after the comma in the last sentence ("to a maximum of 6d8") was added in the 2018 PHB errata, and now makes it perfectly clear how the 1d8 extra damage against undead and fiends interacts with the 5d8 damage cap - the damage cap is increased by 1d8 as well.
Jeremy Crawford explains the reason for the change in the November 27, 2018 episode of Dragon+ (relevant segment starts around 23:52 into the episode). He describes it as a clarification to the feature that matches how many groups were interpreting it, not a change in how it works.
(As clarified in my answer to the Improved Divine Smite Differentiation question, the Improved Divine Smite 11th-level feature adds an extra 1d8 radiant damage, not constrained by this 5d8 limit. In other words, Improved Divine Smite always adds an extra 1d8 radiant damage to qualifying attacks, separately from any spell slots expended to use the regular Divine Smite feature. Crawford explains in the Dragon+ episode that they removed the last sentence of Improved Divine Smite because it was causing people to mistakenly assume it was constrained by Divine Smite's limit.)
Pre-errata, rules designer Jeremy Crawford answered this question (as asked by you!) in a January 2016 tweet:
Does the 5d8 damage cap on Divine Smite count the 2d8 base damage, JUST the extra 1d8 per slot > 1st, or all of it 2gether?
Still unsure what the mac allowed damage of Divine Smite as a whole is. 5d8 using a 4th level slot? or 7d8 using a 6th?
Divine Smite can deal a maximum of 5d8 radiant damage, or 6d8 if the target is an undead or a fiend.
As clearly stated by Crawford, Divine Smite has a regular maximum of 5d8 radiant damage - but that maximum increases to 6d8 if the target is an undead or fiend (since Divine Smite automatically does an extra 1d8 radiant damage when used if the target is an undead or fiend). This was ambiguous before the errata, but is now explicitly stated in the feature description.