Is there a specific group of Adjectives that are used in Dungeons and Dragons to describe spells that get progressively better?

The usage of these modifiers has always been ad hoc

There are patterns, but they aren’t officially defined or anything. Most of them just come naturally out of the English meanings of the words.

D&D 3.5e

The most notable usage of these adjectives are in the “X,” “Improved X,” “Greater X,” “Perfect X” series of feats—Two-Weapon Fighting being the archetypal example. Few chains actually reach four feats long, though, so the “Perfect” modifier is rare (e.g. Perfect Two-Weapon Fighting is an epic feat). When chains are shorter, you see some inconsistencies—for example, you have Weapon Focus and Greater Weapon Focus (and “Improved Weapon Focus” doesn’t exist), but then you have Precise Shot and Improved Precise Shot (and “Greater Precise Shot” doesn’t exist).

There is no commonality between different chains using the same modifiers—while a Greater X will always be a higher-level option than Improved X, neither Improved X nor Greater X has any particular relationship to an Improved Y or Greater Y. A Greater Y feat could easily be available earlier than an Improved X feat and it wouldn’t be notable. (None of these patterns are codified as rules, so it’s not like you couldn’t do Greater before Improved, but it’d be inconsistent with the pattern and therefore a poor choice. To my knowledge, Wizards of the Coast never did that.)

D&D 3.5e also had numerous other “chains” of this sort with other sets of modifiers. Cure wounds and inflict wounds came in minor, light, moderate, serious, critical variants, as well as mass versions of each of those. Least, lesser, and greater appear in numerous other spells. Again, there is no consistency in the use of these adjectives across separate families—within a family “least” would be lower-level than “lesser” but one family’s “lesser” entry could easily be higher-level than another family’s “greater.” There were also plenty of one-off modifiers of a similar sort that we don’t see repeated elsewhere—limited wish being the prime example, in my mind. “Limited” is being used in a similar way but no other spell uses that.

We also see plenty of “families” or “chains” with clear progressions, that nonetheless don’t use any adjective list like this. For example, there is a series of transmutations called bite of the were-X, where X goes rat (1st-level), wolf (3rd-level), boar (4th-level), tiger (5th-level), bear (6th-level).¹ That progression just reflects the power-level of the named creatures.

  1. Spell levels here are given for the druid class; sorcerers and wizards had access to the same spells but each was considered one spell level higher for those classes.

D&D 4e

Some D&D 4e powers were named after spells and other options from previous editions that had used such adjectives, and maintained them. Cure light wounds et al., for instance. Again, there was no cross-family consistency here. Most 4e powers also simply didn’t use adjectives like these, and tended to have more descriptive names.

D&D 5e

Feat chains aren’t a thing in D&D 5e, so you don’t have the same ordering as you did with D&D 3.5e’s feats, and thanks to up-casting, most spell “families” from previous editions of D&D simply become one spell that improves as you use higher-level spell slots (a style reminiscent of the “augmentation” of psionic powers in D&D 3.5e). Nonetheless, you do still see some of these adjectives, used in much the same way as before.