Note that this is subtly different from the “one level in every class” thing; this is about taking your levels one at a time, starting from level 1, rather than a fully-finished level 13 concept.
Could this be at all viable? What order would you go with?
This is just terrible.
I could maybe see a reason to stack multiple spellcasting classes, e.g. Sorc 1 (Con saves, spell slots, Subclass feature, Silvery Barbs) -> Order Cleric 1 (Heavy Armor, Voice of Authority, more spell slots) -> Wizard 1 (Rituals, more spell slots) -> Warlock 1 (Subclass feature, short rest pact slot, Armor of Agathys) -> Bard 1 (more spell slots) -> Druid 1 (more spell slots) -> Artificer 1 (more spell slots). That gets you to be a 6th level spellcaster and 1st level warlock in terms of spell slots, which you spend on things like Healing Word, Bless, Silvery Barbs, etc to proc Voice of Authority. Out of combat you cast a bunch of 1st level rituals and pretty much all the cantrips. Go with a +4 race that nets you 25ft move speed in heavy armor (Fleet of Foot Wood Half Elf or Mountain Dwarf), upcast Armor of Agathys to 3rd level, and go wade into melee, being obnoxious and begging to get hit.
Make sure you die before you hit level 8, b/c the martial classes offer you nothing with only 1 level each. If you do somehow survive and have to take them anyway, go for Rogue & Fighter first (Sneak attack and fighting style, probably Archery, to sometimes be able to hit with a ranged weapon), then Barb (big hit die, since you’re committed to not dying), Ranger (Expertise), Paladin, and finally Monk (the saddest of all capstones).