Enlarge/Reduce allows me to reduce the size of an object by one category (e.g. Gargantuan to Huge) and Animate Object works on objects up to Huge.
If I find an object that's Gargantuan, could two wizards combine those spells (since both require Concentration) to animate and move a Gargantuan object?
If I read it correctly, it would allow the object to move up to 600 feet in any direction during the minute those spells are active.
The heart of the question seems to be: if you cast reduce on an object, does it become a "magical object" for purposes of animate objects?
If the object is a "magical object" then it's an invalid target and the spell fails:
Objects come to life at your command. Choose up to ten nonmagical objects within range that are not being worn or carried.
Unfortunately the rulebook does not define "nonmagical object", so your DM will need to make a ruling.
Your DM might choose to rule that this doesn't work, on the grounds that the phrase "nonmagical object" was probably intended to prevent exactly this sort of trickery. Or they might choose to rule that it does work, because moving a tower sounds pretty awesome and they want to encourage creative use of the rules.
Sage Advice Compendium has some text about whether things are "magical". It asks: (1) Is it a magic item? (2) Is it a spell? Or does it let you create the effects of a spell that’s mentioned in its description? (3) Is it a spell attack? (4) Is it fueled by the use of spell slots? (5) Does its description say it’s magical?
The answer to these questions is "technically no", so we would have to conclude that Sage Advice Compendium thinks the object is nonmagical.
You've clarified in the comments that your goal is to allow an NPC to move a tower onto a small island. Jack points out this quote from the DMG:
For the purpose of these rules, an object is a discrete, inanimate item like a window, door, sword, book, table, chair, or stone, not a building or a vehicle that is composed of many other objects.
so, Rules As Written, you cannot use Reduce on a building.
You might consider the Instant Fortress magic item as a different sort of workaround.
As a bonus action, you can mentally command any creature you made with this spell if the creature is within 500 feet of you
An animated object is a construct with AC, hit points, attacks, Strength, and Dexterity determine by its size. Its Constitution is 10 and its Intelligence and Wisdom are 3, and its Charisma is 1. Its speed is 30 feet, if the objects lack legs or other appendages it can use for locomotion, it instead has a flying speed of 30 feet and can hover. If the object is securely attached to a surface or larger object, such as a chain bolted to a wall, its speed is 0.
No matter what size, your animated object has a base move of 30ft, and has an action to spend as directed which does not restrict it from a "Dash". (Move+Dash)x1min (10 Rounds) would be 600ft. So long as the object you animated was not affixed to something else, in which case it would have no movement speed (e.g. the foundation of a building), and as long as your wizard(s) travel at least a hundred feet with it to stay within the 500ft command zone.
You can't animate any object larger than Huge. Each target animates and becomes a creature under your control until the spell ends or until reduced to 0 hit points.
Regardless of the chicken or the egg (reduce on a Gargantuan or enlarge on a Huge), the target must be Huge at the time of Animate Objects casting. After that, there is no specific mention about changing size breaking the spell.
You cause a creature or an object you can see within range to grow larger or smaller for the duration. Choose either a creature or an object that is neither worn nor carried. If the target is unwilling, it can make a Constitution saving throw. On a success, the spell has no effect.
If the target is a creature, everything it is wearing and carrying changes size with it. Any item dropped by an affected creature returns to normal size at once.
Combining Magical Effects
The effects of different spells add together while the durations of those spells overlap.
The main catch is whether you consider the animated object as (un)willing. Its CON is only 10, and no saves are marked as "proficient" or otherwise boosted so rolling the minimum of DC13 is against odds (assuming only a +1 spellcast bonus minimum).
Beyond that, the order of spells is important for size (what size gargantuan was it before halved / what size huge before doubled), and if it loses STR disadvantage or gains STR advantage. Otherwise you're piloting a gargantuan creature while the spell(s) last and the base stats will be the same with or without Enlarge/Reduce.
Probably not. Animate Object says:
Choose up to ten nonmagical objects within range that are not being worn or carried.
A magically shrunken object is likely a magical object (and would certainly show up to Detect Magic) that is not a valid target for Animate Objects.
