The "Your Spellbook" sidebar for the wizard includes the following line:
Copying a Spell into the Book. When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it.
Does this mean that a Wizard 1/Cleric 19 can copy any wizard spell they find into their spellbook?
Is there anything preventing them from casting these spells?
I would say no to both copying and casting Wizard spells above level 1, based on the spellcasting rules under multiclassing on pg 164 of the PHB:
You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class
I read that as saying that you would count as a level 1 wizard, which comes into play under the and can prepare section. A level 1 wizard cannot prepare spells higher than first level. Nor would they be able to scribe them, as spells you are allowed to know are also based on individual single class.
PHB Example with wizard on pg 164 as well:
As a 3rd level wizard, you know three wizard cantrips, and your spellbook contains ten wizards spells, two of which can be 2nd level
The first errata to the PHB clarifies this explicitly:
Your Spellbook (p. 114). The spells copied into a spellbook must be of a spell level the wizard can prepare.
But, you would be able to cast a level 1 wizard spell as a 9th level spell slot, per PHB rules further below:
If you have more than one spellcasting class, this table might give you spell slots of a level that is higher than the spells you know or can prepare. You can use those slots, but only to cast your lower level spells. If a lower level spell that you cast, like burning hands, has an enhanced effect when cast using a higher-level slot, you can use the enhanced effect, even though you don't have any spells of that higher level.
The 5e rules for multiclassing (PHB p. 164) state that your spells known and prepared are based on your class level for each class, but your spell slots are based on your combined levels, which may result in you having spell slots of higher level than any of the spells you know. They provide an example for Ranger 4 / Wizard 3. With respect to Wizard spells known it says:
As a 3rd level wizard, you know three wizard cantrips, and your spellbook contains ten wizard spells, two of which (the two you gained when you reached 3rd level as a wizard) can be 2nd level spells. If your Intelligence is 16, you can prepare six wizard spells from your spellbook.
This combination counts as a 5th level caster for determining spell slots:
you have four 1st level slots, three 2nd level slots, and two 3rd level slots. However, you don't know any 3rd level spells....You can use the spell slots of those [higher levels than you have spells] slots to cast the spells you do know -- and potentially enhance their effects.
The rules for copying a spell into your spellbook (PHB P. 114 sidebar) require you to have spell slots of that level.
When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it.
RAW then, it appears that the character, who does have spell slots for every level, would be permitted to copy any wizard spell she finds into her spellbook. This comes down to whether "for which you have spell slots" above must be interpreted as "for which you would have spell slots if you were a single-class wizard of your wizard level" There is evidence [see below] that it must be interpreted as such for learning and preparing spells, but the multiclass rules are silent on copying spells, so that leaves the scribing answer as a "maybe".
The spell preparation rules in the Wizard section (PHB p. 114) state that you:
choose a number of spells from your spellbook equal to your Intelligence modifier + your wizard level (minimum of one spell). The spells must be of a level for which you have spell slots.
The learning spells section states:
Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook. Each of these spells must be of a level for which you have spell slots.
Given the example in the first quote of which spells the multiclassed wizard will have learned at wizard level 3, it is clear that "for which you have spell slots" in the final quote above must be read as "for which you would have spell slots if you were a single-class wizard of your wizard level".
Thus we must conclude that multiclass rule statement that "You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class," means that we must re-interpret the phrase "for which you have spell slots" in (at least) these two contexts. The multiclass rules remain silent as to whether we must also reinterpret that phrase in other contexts, such as the rule for copying spells.
Short answer: by RAW, yes.
Wizard:
Learning spells:
Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook. Each of these spells must be of a level for which you have spell slots.
For your first level of wizard, you will get a spellbook with 6 lvl 1 wizard spells. Your first wizard level grants you the ability to add 2 spells to your spellbook, when you gain a wizard level. So if you get no wizard level thereafter, you will learn no wizard spells except by copying them. However, if you were a 18 cleric/2 wizard, then your second wizard level would allow you to add two wizard spells of your choice to your spellbook. At that point (with 20 caster levels) your character has level 9 spell slots, as the multiclassing spell slot table shows, so you can add level 9 spells to your spellbook.
Copying spells
When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it.
To know how many spell slots a character with multiple caster classes has, refer to the multiclassing spell slots table. So a 19 cleric/1 wizard could indeed add lvl 9 wizard spells to its spellbook, assuming the deciphering works out.
Preparing spells
choose a number of spells from your spellbook equal to your Intelligence modifier + your wizard level (minimum of one spell). The spells must be of a level for which you have spell slots.
Here comes the limitation: you cannot prepare more spells than your wizard level + INT. For your 19 cleric/1 wizard, that means 1+int modifier spells (minimum 1, should your INT drop below 13, as required for multiclassing, later). Even if you have a hundred spells in your spellbook, you can only prepare a few. You also only enjoy a very limited Arcane Recovery feature, as you only have one wizard level. However, you can still prepare a lvl 9 spell if it is your spellbook. The rule mentions a level for which you have spell slots - by that they mean spell level, not character level or wizard level. The sentence before did explicitly mention wizard level, so they would have written it there too if they intended that.
How does that compare with other spellcasters?
Ranger, Bard, Sorcerer, Fighter:Eldritch Knight, Rogue:Arcane Trickster
At first level (or second level for the Ranger), you get lvl 1 spells from the respective class' spell list. This first level grants you the ability to learn more spells of the class when you subsequently gain an additional level in it. "Each of these spells must be of a level for which you have spell slots". So if you become a bard 1/sorcerer 19, you will have the starting spells for the Bard and nothing more. However, if you take a level of Bard as last level instead, you will have level 9 spells slots (as shown on the multiclassing spell slots table) and therefore can choose to learn a spell from the Bard spell list of level 1 through 9.
No preparation is needed, if you know spells they are also available to cast.
Arcane Trickster and Eldritch Knight have some additional limitations on choices of spells learned, when you gain levels in those classes. Otherwise, they just require you to have spell slots of the corresponding level.
The Bard's Arcane Secrets are unaffected by multiclassing.
Cleric, Druid, Paladin
You prepare the list of spells that are available for you to cast, choosing from the cleric spell list. The spells must be of a level for which you have spell slots.
No learning spells is needed, you have total access to all spells on your class list.
However, like the Wizard, you do need to prepare them to make them available for casting, and are limited to a number equal to your cleric levels + your wisdom modifier. Additionally, "the spells must be of a level for which you have spell slots". As a Wizard 19/cleric 1 character you would be able to prepare 1+wis mod spells, that you can choose from the complete cleric list, since you have lvl 9 spell slots. This is not affected by the timing of your cleric level: at any point, you can prepare any cleric spell if you have spell slots of that spell's level.
Warlock
The Warlock functions separately from other caster levels. Multiclassing is simply adding abilities together.
Some people argue that RAI spells known should be strictly limited to separate caster level spells slots. They're free to houserule so, but RAW is clear as written. There exist no errata or Sage Advice Compendium that contradicts it, only tweets (that have been wrong before) and a version of the compendium that was later pruned.
Clearly, the benefits of multiclassing spellcasters are balanced. Yes, you get a broader spell selection. However, you give up high-level abilities for it. In addition, you lose out on an ability score increase (which is doubly damning because of MADness), and can't enjoy the class-level keyed benefits like sorcery points, and most importantly the number of spells prepared... which strongly limits the number of spells you could use. Most classes only learn 1 or 2 spells of any level at level 2, and for two two spells you have to sacrifice at least the last two highest levels of your character.
"Spells known and Prepared: You determine what spells you know and can prepare for each class individually, as if you were a single-classed member of that class."
I interpret this to mean that I should also be looking at the slots chart of each portion of my multi-class character individually before determining if I can know or prepare a spell of a certain level. In other words...regardless of how many or what level of spell slots my character has access to due to being a multi-classed character, he/she is still limited to the number and levels of slots that they would have had as a single class character. 1st level wizard as per the single class wizard chart has 2 first level spell slots and therefore will have a maximum access to 1st level wizard spells even if his multi-class total caster level is granting him higher level spell slots.
All in all it's still pretty damn sweet that multi-class characters get higher level spell slots that they can use to ramp up the power level of the lower level spells that they actually know and prepare. That in itself is worth celebrating without needing to squeeze more out of it than was likely intended.
Spell Slots. (PHB pg 164)
You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, half your levels (rounded down) in the paladin and ranger classes, and a third of your fighter or rogue levels (rounded down) if you have the Eldritch Knight or the Arcane Trickster feature.
Learning Spells of 1st Level and Higher. (PHB pg 114)
Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook. Each of these spells must be of a level for which you have spell slots, as shown on the Wizard table. On your adventures, you might find other spells that you can add to your spellbook (see the "Your Spellbook" sidebar).
Copying a Spell into the Book. (PHB pg 114)
When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it.
