According to the wording, I would say no, the regeneration effect of Regenerate does not benefit from Disciple of Life.

Disciplie of life says: Whenever you use a spell of 1st level or higher to restore hit points to a creature, the creature regains additional hit points equal to 2 + the spell's level.

So this effect is an additional effect to the spell, does not increment the spell's effect itself.

I would say this:

In case of Goodberry, you cast the spell but the effect (restore 1 hp) triggers when the berries are eaten. When one eats a berry triggers the spell's effect (1hp), plus the additional effect of Disciple of Life.

In case of Regenerate, you cast the spell and immediately one effect is triggered (restore 4d8 + 15). At the same time, the additional effect of Discipline of Life is triggered as well.

So, in my mind, it is a matter of when the restoring effect of a spell are triggered. As soon as this happen, the additional effect of Disciple of Life occurs as well.

I'm going to run through my thinking on this step-by-step (and in real-time, because I don't know my opinion on the matter at this moment during the typing of this post)

There are clauses made for the disciple of life effect:

1. spell of 1st level or higher (check, regenerate is 7th level)
2. restore hit points to a creature (check, regenerate target's a creature and says "regains ... hit point(s)")
3. whenever

The last one of the three clauses is kind of unclear. The "whenever" could be referring to the use (read: casting) of the spell, which would limit disciple of life to one application per casting... or the "whenever" could be referring to how often the effect is allowed to apply, which would limit disciple of life to one application per instance that hit points are regained as a result of the spell.

In the case of goodberry, there is official clarification that disciple of life's "whenever" is being used in the later fashion, since each berry benefits from the increase to number of restored hit points.

That means the most consistent application of rules would be to have the 4d8 + 15 hit point portion of the spell and the 1 hit point at the start of each turn portion of the spell be increased by 9 thanks to disciple of life.

Whether that is intended or not, I can't say. I can say that I don't see a significant enough difference in the language of goodberry and regenerate to consider as evidence of the intent for the two spells to be treated differently.

And I can also say that I don't think the effect of handling the spell as 4d8 + 24 upon casting and 10/round for up to an hour is too potent for the game given the cost being a 7th-level spell slot and a 1 minute casting time  - that casting time means not being able to use it mid-combat, and potentially having its practical effects limited should you misjudge when to cast it or the target not actually end up taking that much damage during the spell's duration. And, of course, at the level of play when the spell becomes available it isn't actually unlikely that some foe you face be capable of dispelling the effect.

I agree with Aaron. If Disciple of Life applies to Goodberry, it should also apply to the HP Regenerate heals every turn.

It's a strong combination, but I don't think it's game-breaking:

• This is about the same amount of regeneration monsters gets. Compare to trolls (CR 5, 10 HP/turn), death slaad (CR 10, 10 HP/turn) and vampires (CR 13, 20 HP/turn).
• It's easily countered with lower level spells. Chill Touch is just a cantrip and it'll stop all healing for one round. There's also Dispel Magic, which every spellcasting class other than Ranger gets. A level 7 slot is worth about twice as much as a level 3 slots (Spell Damage, DMG p.283; Variant: Spell Points, DMG p.288) so even if it takes 2-3 tries, it's still a fair trade.
• There's plenty of spells, monster abilities and conditions that can take the regenerating character out of the fight without reducing their HP to 0.
• Level 6-9 spells are supposed to be super powerful. By the time you get them you're in a different tier of play (PH p.15) and the game limits them severely in terms of spell slots. They're always excluded from class features and magic items that recover spell slots or store spells.
• Since those spell slots are super limited, there's a big opportunity cost to using them. Casting Regenerate means you're missing out on Conjure Celestial, Fire Storm, Divine Word or Etherealness with that spell slot.

Can you clarify if Disciple of life stacks with Preserve Life?

Quote from Willy8wonka1234 >>

Can you clarify if Disciple of life stacks with Preserve Life?

 The two features have no interaction. Disciple of Life augments the healing provided by spells, and Preserve Life is a Channel Divinity option and thus not a spell.

I don't think a 10hp Regen is gamebreaking.

A fighter with 20 CON and the champion archetype has a HP Regen of 10HP per turn when under half health at lvl 18. All the time, not just when a spell is successfully cast and maintained.

And there's probably a bunch of feats (and a fighter can take all of them) which could increase this