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At level 18, Wild Magic sorcerers get the Spell Bombardment feature (PHB, p. 103):

Beginning at 18th level, when you roll damage for a spell and roll the highest number possible on any of the dice, choose one of those dice, roll it again and add that roll to the damage. You can use the feature only once per turn.

When you score a critical hit with an attack (even a spell attack), you double the amount of dice you roll.

Does the additional damage die from Spell Bombardment also get rolled twice on a critical hit?

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Yes

The extra damage from Spell Bombardment is a damage roll and is rolled twice on critical.

This is the same case with adding superiority dice to the damage roll when using a battle maneuver. The wording is similar:

Commander's Strike
[...] adding the superiority die to the attack's damage roll.

Disarming Attack
[...] You add the superiority die to the attack's damage roll, [...]

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The die from Spell Bombardment is a damage die, so it will be doubled on a critical hit

The section on "Critical Hits" states:

When you score a critical hit, you get to roll extra dice for the attack's damage against the target. Roll all of the attack's damage dice twice and add them together. Then add any relevant modifiers as normal. To speed up play, you can roll all the damage dice at once.

Therefore, what we need to determine is whether Spell Bombardment's die counts as a damage die.

"Damage Dice" are never strictly defined by the rules, so they mean their natural English meaning, dice which are rolled that contribute to damage. This answer to the question "How do critical hits and bonus damage interact?" states:

Any damage expressed as dice gets to be rolled twice, but other modifiers are added only once.

Following this, any dice used in calculating damage are "Damage Dice" and so they are doubled on a critical hit.

The die from Spell Bombardment is a die roll that contributes to damage, thus it is a damage die, and so it is doubled on a critical hit.

There is also further support for this found in the Sage Advice Compendium:

Q. If a shadow rolls a critical hit, does it reduce the target’s Strength by 2d4, as well dealing the extra necrotic damage?
A. No. A critical hit lets you roll damage dice twice. An effect that deals damage is one that reduces the target’s hit points. The shadow’s Strength reduction isn’t damage, because it has no effect on the target’s hit points.

Note: Whether or not "negative dice modifiers" count as "damage dice" is discussed in the question "Does the Reduce option from the Enlarge/Reduce spell cause a critical hit to do 2d4 less damage?", but they do agree that "positive dice" count as damage dice.

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