Fighter/Paladin
The most obvious exploit here is that you can combine the fighter's 11th-level damage buff (Extra Attack for three attacks) with the paladin's 11th-level damage buff (Improved Divine Smite for +1d8 radiant damage per attack). This lets you deal the +1d8 radiant damage three times per round, which under the normal rules would not have been possible.
On the other hand, you can't do that until sixteenth level; not many games last until sixteenth level, so this is unlikely to ever come up.
Fighter/Wizard (or Fighter/Bard)
Another potential exploit would be to start with a Bladesinger Wizard, which grants full wizard caster progression and Extra Attack. A wiz6/ftr5 would have spellcasting as a sixth-level wizard and triple-attack; without this rule, the best you could do would be an eleventh-level Eldritch Knight, which is the spellcasting equivalent of a fourth-level wizard. Combining this with (eg) the shadow blade spell could lead to more damage than the designers probably intended.
(Shadow Blade does more damage than most weapons, and part of the balance is that anyone who can cast it probably doesn't get a lot of attacks per round.)
The bard class has a few subclasses that work similarly.
Fighter/Ranger
I don't see any obvious exploits involving multiclass fighter and ranger. The ranger class does get some small damage buffs to make up for it not getting triple-attack at eleventh level, most notably hunter's mark which scales with the number of attacks you make. On the other hand, a fighter who really wanted hunter's mark has other ways to get it.
Sample Build
One possible build would be a Ftr5 / Bladesinger Wizard 6 / Valor Bard 9. This character could cast shadow blade using a seventh-level spell slot, getting a weapon that deals 5d8+DEX on a hit. They qualify for four attacks with their Extra Attack, so that's 4*(5d8+5)=110 psychic damage when they attack. But, also, they have Action Surge, so once per short rest they can do 220 damage.
(For comparison, a 20th-level Eldritch Knight fighter with a +3 greatsword would attack for 4*(2d6+8)=60 damage, or 120 on an action surge. The Eldritch Knight would have access to fourth-level spell slots, compared to this character who has access to eighth-level spell slots.)
It's a bit silly to judge balance based on 20th-level characters, because characters are level 20 for less than 5% of their existence. But, well. This seems too good.
The best argument for not allowing this probably sounds like this: "Levels 6+ of Fighter mostly aren't that exciting. You get a few extra stances, your archetype gets a little better, the Indomitable thing lets you reroll a save once in a while. The only reason anyone would take more than five levels of Fighter is because they want the Extra Attack bonus, which is the single special thing that only fighters get. If you tell your players that you're giving that bonus to multiclasses, then it's a no-brainer to multiclass."
And, also: "The benefit of multiclassing is you get a bunch of shiny new abilities right away, like spellcasting and smites and bardsong. The cost is that all the high-powered stuff is buried deep in a single class's progression, and if you multiclass then you're not going to get it. If you issue this houserule then you're taking away the main drawback of multiclassing."
The main argument for allowing it is that getting Extra Attack a second time is a dead level, and that's kind of a bad feeling. Getting Extra Attack a second time should get you something, shouldn't it?
Another argument for allowing it is that many campaigns don't actually last until eleventh level (eg, most campaign books don't run until level 20), so if this does cause a problem, your game might end before you notice it.
Overall, my advice is to rule no, if you think there's a significant chance of your game running past eleventh level.
