Finding The Greatest Steed! (5e Optimized Character Build)
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Slow And Steedy
If you want the strongest possible mount build in D&D 5e, you've come to the right place. This build makes the assumption that all you care about is having the most powerful mount, with being a powerful character yourself coming second. It's still important, but not as relevant to our cause. After all, having a powerful mount means nothing if your character bites the dust, kicks the bucket, or worse, dies; so we'll try to make the most of our build as a character and a mount, but the mount will come first.
Step 1: Half Elf Bard. Other races might be optional depending on your ability scores, but the extra skills and ability score increases from Half Elf pair really well with what we are doing. You will want Bard as your main class, but we only need 18 levels of Bard to get what we need from it. Half Elf Bards do get a lot of skills, but the one we need the most is Arcana (Probably, ask the DM what knowledge skill would be needed to know things about planar creatures).
The Thick Of It
The main strategy of this build is to take Lore Bard as your subclass and abuse the Magical Secrets feature to its fullest extent. Get Expertise in Arcana at level 3 to get a head-start on being the lore researcher that you need to be. You're looking for a specific fact of knowledge and your character doesn't get to know that by default. Make their interests align with knowing about Aberrations and Dragons (Or just other planar creatures broadly, depending on how good your DM is at figuring you out). Take Haste for your Additional Magical Secrets as a level 6 Lore Bard, and take Find Greater Steed for your Magical Secrets feature at level 10. Then take True Polymorph as your 9th-level spell for level 17. Everything else is optional, but you'll want to capitalize on the combo so I'll add extra combo potential in the next section.
In the meantime, you'll use this packet of powers to True Polymorph your Griffon or Pegasus into an object of any kind. I recommend something that won't easily be destroyed. Then take that object/steed and True Polymorph your pet rock into a CR 9 creature as per the spell (If you're clever, you might have thought of a weakness of the build already. I'll detail those at the end of the build). What most players do is make it a Young Silver Dragon so they can ride a dragon. Good. Cool. WEAK. We can go further. Instead of doing that, turn your pet rock into a CR 9 Gray Slaad. The specific fact you're looking for that you hopefully found within 17-20 levels of play/ A Gray Slaad that eats the entire corpse of a Death Slaad instantly transforms into a Death Slaad. You just upped the max CR creature you can True Polymorph your steed into from measly 9 to powerful 10; The same CR as the coveted Young Gold/Red Dragon. Keep in mind, throughout this process the creature is technically still exists from Find Greater Steed. Now that's a lot of hit points! Combine that with the fire immunity, and you have quite a tough mount.
Grab Bag Of Awesome Combos
You might have noticed I left a lot out of that previous section. That's the main strategy of the build, but you can go even further beyond. Spells you cast on yourself also affect your steed while you are riding on it, and some feats and features from elsewhere can also combo with the build in interesting ways. You have 8 Magical Secrets after all, not 2. For most of the game your steed will be stuck as a Griffon or Pegasus, and that'll happen again if your final product is ever bested or some meanie casts Dispel Magic at 9th-level on it. The above combo probably can't be done more than once without the DM actively stopping you from doing it again, but you have a long way to go to make it that far. Until then, there are other ways to make your steed a powerful, relevant force that the party can rely on.
A Griffon or Pegasus both fly, and both flying speeds (As well as your dragon's flying speed) are affected by the Haste spell (usually). They are frontline fighting creatures with no range and low AC and hit points (For late game), so the extra AC and extra attack are useful as well. There's also a feat that is a must have for this build: Inspiring Leader. Up to ~25 temp HP for you and your mount that isn't a product of spells means your mount can take an extra hit or two, and since it affects 6 creatures, a small party can also gain the full benefit and be much tougher. Crown of Stars can also be doubled, giving your mount a bonus action RANGED attack that it wouldn't have otherwise while letting you have one as well. The Greater Invisibility spell and Polymorph spell also works with it (Though until you do the thing your mount is limited to CR 2 forms). The Enhance Ability spell also works too. If it has a form that can communicate, the Glibness spell can also work on it. Grab Wish as your 18th-level Magical Secrets spell just in case you think of another combo (And because Wish is top tier). The Mounted Combatant feat is a given, but not necessary compared to the Inspiring Leader feat and War Caster feat if you want better ability scores. Finally, and most importantly when you don't have a powerful mount yet, healing spells you cast on yourself can also target your mount, such as Healing Word and Cure Wounds. Also, grab Feather Fall at 1st-level. You'll want it at some point since your mount options can all fly but YOU can't.
Besides spells and feats, you only need 18 Bard levels, so you have 2 extra to do what you want with. You could get Fighter for the Protection Fighting Style and make it harder to attack your mount. If you do so, you will want the War Caster feat to cast spells while mounted and wielding a shield. Alternatively, you could get two levels of Wizard to get Divination Wizard's Portent, to potentially affect rolls that could help or harm you or your mount. Or get two Warlock levels for Eldritch Blast, invocations, and massive sniping potential (You can fly and snipe). Other class feature combos likely exist, but I can't think of everything so that's where you can add your personal flair to it. Alternatively, taking 2 levels in something other than Bard makes you lose your last ability score improvement, so going level 20 Bard is a legitimate strategy as well.
Your character isn't exactly a slouch as an 18th-level full spellcaster and Lore Bard. You still get all of its powers on top of your exploit, which includes extra skill proficiencies from being a Half Elf and from your background to have some skill monkeying around, including Jack of All Trades for anything you're not proficient in. Expertise also still exists (You only need Arcana probably, so you can put the other three anywhere else you'd like), and you could maybe grab the Prodigy feat for even more of all of this (At the potential cost of missing out on something else important). Your Lore Bard's Cutting Words feature can also help protect you and your mount. Finally, we didn't need all of your Magical Secrets, so you have some extra spells from any spell list in the game you can use to augment your combat or out-of-combat abilities.
The Catch And The Limit
A lot of things have to go right for this ideal set up. You have to get the DM to interpret the Find Greater Steed spell AND the True Polymorph spell a specific way. You have to survive that long with a build that picks up steam at level 10. Most games don't reach level 17, preventing this combo from even potentially happening. You then have to kill a Death Slaad without making the rest come after you, AND the DM has to interpret that your fake Gray Slaad can transform in this way. Oh yeah, and you have to get that info in-character in the first place, something which may never happen depending on the circumstances of the game. If you plan to play this build, I recommend you talk to the DM about it first, to make sure you even have that possibility of making things work out this way.
Additionally, there is at least one spell that shouldn't be used with this build but seems to feel like it could be: Tenser's Transformation. At first glance, this spell is perfect for your mount. It isn't; Your Inspiring Leader feat will give temporary HP (The spell gives more but they don't stack and your feat's temp HP doesn't end when the spell ends) and most of the other features such as the Extra Attack feature don't stack with the Griffon's attacks (Does work with the Pegasus though.) The advantage on weapon attacks and the extra force damage is all your mount really gets, and as a Concentration spell you getting hit could end it at any time. You'll want high Constitution to avoid that, but there's no guarantee. But you as a character won't be using weapon attacks (You have spells), which means it really doesn't double the spell benefits like other spells (Such as Haste) do.
Finally, obviously the biggest limitation is the set up itself. You will probably never make the core Young Gold/Red Dragon transformation happen twice in one campaign. Even if you do it once, you now have to contend with the rest of Slaad-kind for your trespass. The Death Slaad is also chaotic evil aligned, so you'll have to contend with the alignment shift as well during the transformation process. You only have one 9th-level spellslot until you start getting Boons at 20th-level, so most likely you'll be doing this process one day at a time, during probably the climax of the campaign or the road leading up to it. Additionally, none of it matters because while Find Greater Steed won't be dispelled (The spell ends when you summon the mount, so there isn't a spell to dispel with Dispel Magic) the True Polymorph explicitly can still be dispelled even after becoming otherwise permanent, so a caster final boss will almost certainly nerf your mount back down to normal. There's also the whole reason this build exists in the first place. In order to do ANY of this, you need the spell True Polymorph. But to benefit from an awesome mount in this way, you can't use True Polymorph on yourself to turn into anything larger than a medium-size creature. Which you could otherwise use to, you know, turn into an ANCIENT dragon (You can however use True Polymorph to turn yourself into a Red Abashai or a Kalaraq Quori and still have your mount though.).
This is an incredibly tricky build to pull off, but to my knowledge it's the BEST possible mount build.