If I'm picking a single classed character, but it's against the rules to pick any full caster - well, that right there shows that fully 6 of the 12 classes (the full casters, including warlock) can't be done better with multiclassing.
I was really only speaking of the Wizard and Sorcerer - and a few types of druid, cleric and bard. The ones that rely upon spells first and foremost and for which weapons, etc... have no value.
Okay, so let me pick among the remaining classes. Let's go simple: A front line melee tank/striker/support - the Oath of Ancients Paladin. Features I'm looking at:...
...looking at more than a role. You're specifying minor details. Here are the role relevant concepts from your description: Melee Jack of all Trades capable of high damage, durability, and support (healing and protecting allies). The methods by which they meet these goals don't need to be identical to the pure Ancient Paladin.
I would recommend that the multi-class take place at level 9. We want that level 7 ability of the OotA Paladin and the ability score increase at 8.
Sorcerer is a great option to start combining with the Paladin. Let's just assume we're adding 12 Sorcerer levels - and to stay in theme, Divine Soul so we have access to cleric spells with a high degree of overlap with Paladin spell lists. Warlock also has potential, and electing to take one level of Hexblade to get Charisma based attacks is a nice benefit, but as the character can afford to have low (Dex, Wis and Int), you do not need to get that Charisma to attacks benefit. You can start out with a 16 Strength and then improve it via magic items or ASIs... but you might keep it parked at 16 for quite a while. I would consider adding in 4 Warlock levels starting at 13th or 17th level.
Incidentally, I did run a 20th level Paladin/Sorcerer/Warlock combo in a three session one shot and it was very strong, but it was 14th level Vengeance paladin/3rd Hexblade Warlock/3rd Dragon Sorcerer - a combo that made sense as an endpoint but that would have been highly inefficient for many levels. Still, it did a lot of damage.
So what are we losing by multiclassing? Lay on Hands advancement (5/lvl), Higher level Paladin Spells (3rd at 9, 4th at 13, 5th at 17), Aura of Courage (10), Improved Divine Smite (11), Cleansing Touch (14), Undying Sentinel (15), Improved Aura (18) and Elder Champion (20).
Lay on Hands: You'll have less healing to offer - but you'll have enough for the emergency purposes for which LoH is often used. You can use some of the extra slots you get to cast healing spells if needed, but the real way that you'll make up for this is by using spells and abilities to better avoid damage in the first place. Once you hit 14, Empowered Healing will help here.
Spellcasting: We're going to gain higher level spell slots faster from level 10 and on, but not have actual higher level spells to put into them for a bit. We'll get to 5th level spells under both approaches at 17th character level... and both will have access to 4th level slots at 15th character level. At that 15th level, the multiclass character has 2 additional 4th level, 1 5th and 1 6th level slot... which can be huge. Cast banishment on 3 enemies using quicken and then beat up the one that stays there?
You can dig into the minutia as to what we can do with the spell slots here, but you're trading access to a very small number of higher level paladin spells for a greater number of prepared spells that come from low level paladin and cleric/sorcerer, a greater number of spell slots overall, and a wider range of spells available. Given that you can use those extra spell slots to smite, it seems like a net gain to me. We're also going to get some metamagic here as well so there is a lot of spellcasting advantages here - the main benefit of the multiclass.
Courage Aura: This has always been a rarely used ability - and you're already giving save bonuses. It becomes more valuable at the highest levels, but losing it is not much of a loss. The other gains from being a sorcerer make up for it. Favored by the Gods helps you avoid the fear. Spells can remove it from allies.
Improved Divine Smite: This is a huge loss - but you're getting Metamagic in place of the extra d8/attack and d8/smite. You can use that to cast quickened buff spells and quickened debuff spells in addition to your attacks. You should be able to make up for that damage loss with the extra spells.
Cleansing Touch: You'll have to remove those spells using dispel magic and your slots. Favored by the Gods will help you avoid the spells.
Undying Sentinel: Those additional spells should help you stay on your feet and not need this benefit.
Improved Aura Radius: By now your allies are good at staying near you. It would be nice to offer the protections at greater range, but those higher level spell slots (7th level slots!) are nice.
Elder Chump: Your Capstone - You essentially get a one round spell that does not require concentration, debuffs enemies, allows you to quicken all spells for no cost, and regenerates 100 hps. Your 6th level spell makes up for not getting this ability. Mass Suggestion also has a massive impact and does not require concentration.
If I'm trying to build a character that defends myself and allies while hitting hard withstanding punishment, I feel like this soradin does it better than a pure OotA paladin in most games. However, if you have a *lot* of encounters per rest, the soradin's spell slots/spell points might not last as long as we'd like - the paladin abilities stretch a bit better over more encounters.
I also think a good Warlock/Paladin with 12 levels of Warlock could also outperform the pure OotA Paladin, but I won't go through that build entirely. Another good build would be adding 3 levels of sorcerer following level 12 as a paladin, and then following that with 5 levels of warlock.
