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The Paladin's Divine Smite feature description states:

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.

Since it doesn't say "Once per turn" or something along those lines, could I just expend all of my spell slots to do a crap ton of damage?

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What do you consider "smiting"?

Paladins have a number of ways of adding damage to their melee attacks that include the word "smite" in their name. However, none of those things actually interact with each other, so someone might reasonably argue that each of them is their own separate thing, and maybe only the Divine Smite class feature really counts as smiting.

As you quoted the Divine Smite class feature in your question I'll assume that's the main aspect of smiting you're interested in, and expand from there.

Divine Smite caps at 5d8 or 6d8 per hit

Note the section you quoted: "... you can expend one spell slot..." (emphasis mine). If you could expend one or more spell slots, it would use the plural for spell slots. This gives a general cap of 5d8 per hit, with an extra 1d8 per hit against fiends & undead.

88 or 112 damage from just Divine Smite

If you're purely interested in spell-slot smiting and don't factor in anything else with the word "smite" in it, then a multiclass level 11 fighter and level 9 paladin has 3 attacks from fighter Extra Attack and has two level 3 spell slots they can use for smiting (the third attack has to settle for using a level 2 spell slot).

That gives us 4d8 + 4d8 + 3d8 = 11d8 for a maximum of 88 damage. Against fiends or undead we get an additional 1d8 per hit, pushing us up to 112 damage.

96 or 112 damage with Improved Smite

The level 11 paladin feature Improved Divine Smite adds 1d8 damage to all melee attacks.

A level 15 paladin has Extra Attack allowing them to attack twice in a single action, Improved Smite for an extra 1d8 per hit, and two level 4 spell slots they can expend to boost those hits.

That gives us 5d8 + 1d8 + 5d8 + 1d8 = 12d8 for a maximum of 96 damage. Against undead or fiends we get an additional 1d8 per hit, pushing us up to 112 damage.

146 or 162 damage with smite spells

A level 17 paladin could use Extra Attack to attack twice in a single action, expend a level 4 spell slot for Divine Smite with each attack and also cast Banishing Smite on one of those attacks (the limitation on casting multiple leveled spells in a single turn prevents them from casting it on both attacks), plus the previously mentioned Improved Smite.

That gives us 5d8 + 5d10 + 1d8 + 5d8 + 1d8 = 5d10 + 12d8 for a max of 146 damage. Against fiends or undead we get an additional 1d8 per hit, pushing us up to 162 damage.

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A maximum of 96 (12d8), or 112 (14d8) against undead and fiends

The description of the feature says (emphasis mine):

[...] 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.

And there is no limitation on the number of times you can do it in a turn, provided you hit the target. Beware that you can spend one spell slot per hit. Hence, if you have the Extra Attack feature, which allows to attack twice in a turn, you can use your smite twice in a turn.

Compare it with the Rogue's Sneak Attack feature:

Beginning at 1st level, you know how to strike subtly and exploit a foe’s distraction. Once per turn, you can deal an extra 1d6 damage to one creature you hit with an attack if you have advantage on the attack roll.

Obviously, the maximum number of d8s to roll for the extra radiant damage depends on the slot level used, hence on the paladin's level. Considering a 15th level paladin, we have Extra Attack Feature and Improved Smite, the latter giving a further 1d8 of radiant damage on a hit (without spending a slot): this allows to expend two 4th level spell slots, each providing 5d8 radiant damage, plus 1d8 radiant damage given by Improved Smite for each hit. The grand total is then $$ 12\mbox{d}8=5\mbox{d}8 + 5\mbox{d}8 + 1\mbox{d}8 + 1\mbox{d}8 $$
This becomes 14d8 when the target is an undead or a fiend.

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