I can't find a reference within the core books that states how many spells can be cast in a round. I do see how many segments it takes per spell. Can a wizard cast multiple 1 segment spells i.e. magic missiles in a round given that they roll an initiative less than 8 and have two copies of the spell prepared?
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Casting a spell takes a magic-user's entire round. They may not take any other actions, including moving. The number of segments listed for a spell is only used to figure out when during a round the spell completes, which can be important for figuring out whether the spell is interrupted (and spoiled).
For a complete breakdown of the spellcasting process during combat, see page 65 of the DMG.
Learning AD&D for the first time from the original books takes a ridiculous amount of dedication due to the bizarre organisation, the rambling discoursive style, and the way a single rule can be spread across multiple chapters or even across the three core books. I certainly didn't learn the fundaments that way – like most AD&D players, I learned by playing with people who already knew how to play.
To learn directly from the books requires dedicating a lot of time, and using that time to read and re-read them, from cover to cover and in bits and pieces, while putting them to use in an ongoing campaign. A large tolerance for doing it "wrong" for a good while is helpful.
As an alternative, I can recommend reading OSRIC, which is a retro clone of AD&D. Its advantages include being more concise (it's only about 40 pages of rules plus 80 pages of spells), organised more like how we expect rules to be organised today, and free. It makes a good companion to the original books, and can help clarify rules confusion since it's much easier to find everything.
The conciseness of OSRIC is also its downside: it lacks the rambling commentary that puts the rules in context, and which sometimes provide more nuanced guidance that falls somewhere between "background" and "rules". For example, OSRIC makes no mention of spellcasting preventing crouching when a spell has somatic components. I never knew about that rule(?) until I re-read the several different sections in the PHB and DMG about spellcasting while I was looking for a citation for this answer.
For a complete breakdown of the RAW combat rules for first edition AD&D download the ADDICT PDF from Dragonsfoot.
The short answer is one spell (DMG Page 65) or two cantrips (UA Page 45).
Both side's spell casters indicate which spell they are intending to cast PRIOR to the roll of the initiative die. (DMG Page 65)
During surprise combat, the spell caster can begin casting a one-segment spell, or begin casting a longer spell. (DMG Page 61)
The following procedure is found on Pages 7 to 8 of ADDICT. I omitted the examples but included the page references.
Spell Casting: Even when a spell caster has won the initiative, it is possible for an opponent’s attack to occur before the completion of (and thus disrupt) the spell. (DMG Page 66-67)
_1. If combat is spell vs. spell, the spell with the shorter casting time will be completed first, with the initiative result breaking ties in casting times.
_2. If the spell caster is being attacked by a melee weapon with a speed factor rating, use the following procedure to determine when the weapon strikes.
__a. When the attacker wins initiative, the weapon will always strike prior to spell completion.
__b. When initiative is tied, a straight comparison of weapon speed factor to spell casting time will determine which occurs first. Simultaneous results are possible.107
__c. When the attacker loses initiative, subtract the attacker’s losing initiative die from the weapon speed factor (treating negative numbers as positive), and compare the result to the spell casting time to determine which occurs first. Again, simultaneous results are possible.
_3. If the spell caster is being attacked by missile weapons, or by natural weaponry without a speed factor rating (such as a monster’s claw/claw/bite), use the following procedure to determine when the attack strikes.
__a. When the attacker wins initiative, the attack will always strike prior to spell completion.
__b. When the attacker loses initiative, or when initiative is tied, the attack will occur on the segment indicated by the caster’s initiative die. Compare this to the spell casting time to determine which occurs first. Simultaneous results are possible. (DMG Page 65)
_4. When the attacker has multiple attack routines, only the first attack has an opportunity to disrupt a spell (unless the spell requires a full round to cast).
_5. A magic-user may cast two cantrips in a single round. (UA Page 45) In such a case, the timing of the first cantrip is determined as above. The second cantrip will be cast 1-4 segments later.
_6. The spell caster cannot use his or her dexterity bonus to avoid being hit during spell casting; doing so interrupts the spell (DMG Page 65)
_7. Any successful attack, or non-saved-against attack, upon the spell caster interrupts the spell (DMG Page 65)
_8. The above procedures for resolving the timing of attacks against spell casters also apply to opponents using magical devices with specific activation times (such as rods, staves, and wands). These functions, however, will not be automatically disrupted by a successful attack (unless the wielder is slain or otherwise incapacitated).
In Original Edition:
V3 p.9, via implication:
Chainmail (which is intended by the D&D rules text to be available as part of the core, even tho' it wasn't used for combat in EGG's games by 1975) includes on p. 32:
It depends on your point of view.
If you're measuring the wizard's time it's 1 spell as others have said.
However, if you're measuring world time there is no limit. 1st edition spells don't have caps on their effects and for every 20 levels you'll get 1 round in Time Stop. A 200th level wizard thus could cast 11 spells in a round--the Time Stop and then 10 while inside it. (Note that anything instantaneous can only affect him.)
Finding the 200th level wizard I'll leave as an exercise for the reader...