In our game, the arcane caster went on hold to counterspell Teleport of a demon to stop it escaping at the last moment (unknown to the player the demon had the greater version).
Can you counterspell with the lesser versions of spells?
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The rules are quite clear about this. Allow me to quote from the rules for counterspells found on page 170 of the player's handbook:
As a general rule, a spell can only counter itself. For example, a fireball spell is effective as a counter to another fireball spell, but not to any other spell, no matter how similar. Fireball cannot counter delayed blast fireball or vice versa.
Greater teleport, like delayed blast fireball, is a more powerful version of another spell. In fact, both even use the exact same "This spell functions like X, except..." wording. There is therefore no reason to believe that greater teleport can be countered by teleport, any more than delayed blast fireball can be countered by fireball.
In addition, even if you could counterspell greater teleport with teleport, you can't counterspell a demon's innate greater teleport spell-like ability because spell-like abilities cannot be counterspelled. The relevant rule is on page 180 of the Player's Handbook, under the "Special Abilities" heading:
Spell-like abilities cannot be used to counterspell, nor can they be counterspelled.
No, not even with the Improved Counterspell feat (Player's Handbook 95) which also summarizes the normal counterspell rules:
Benefit: When counterspelling, you may use a spell of the same school that is one or more spell levels higher than the target spell.
Normal: Without this feat, you may counter a spell only with the same spell or with a spell specifically designated as countering the target spell.
The arcane caster could have readied to dispel magic (PH 223) in an attempt to obtain the desired result, though.
The rule for this is
It is possible to cast any spell as a counterspell. By doing so, you are using the spell’s energy to disrupt the casting of the same spell by another character. Counterspelling works even if one spell is divine and the other arcane. (...) As a general rule, a spell can only counter itself.
A spell of closely related function, even if more powerful, is not the same spell.
That being said, if you and your DM agree that it would make sense flavour-wise, it of course is entirely up to you to rule that way.