We were playing against a boss which could cast Deeper Darkness at will. If we had Daylight (which we didn't) or Dispel Magic (which we didn't) and countered his casting, would he be able to just cast it again on his next round?
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There isn’t really any such thing as at-will spells, and you cannot counterspell SLAs
Those were probably spell-like abilities (SLAs for short). You cannot counterspell spell-like abilities at all (nor can they be used to counterspell).1
When you do counterspell, the slot is gone but they can still cast the spell if they have more
A successful counterspell against an actual spell does burn the spell slot. If the target is a prepared caster like a wizard, and only prepared one, they would not be able to cast it again. A spontaneous caster like a sorcerer could, but would be down a spell slot (and if that happened to be his last slot of that level and he had none of a higher level, he also would be unable to cast it again).
Instead of counterspelling, counter-nuke
However, there is a better option than counterspelling, and it does work against spell-like abilities (and you can use spell-like abilities to do it; for that matter, you could use a bow and arrows). Instead of readying an action to counterspell, ready an action to nuke the target as soon as he starts casting. When someone casting a spell or spell-like ability is damaged, they must make a Concentration check to avoid losing the spell. If you hit him hard enough, that Concentration check will be much more difficult than resisting dispel magic, plus you also get in damage as well as burning the spell. Reliable high-damage ranged touch attack spells, like orb of force (Spell Compendium), are ideal for this purpose, since touch attacks are usually very easy.
Note that this tactic is extremely brutal, and using it may trigger your DM to do the same, which may be unpleasant.
1 If you have dispel magic or greater dispel magic as spell-like abilities, those may be able to counterspell, depending on interpretation. The caster’s lament warlock invocation (Complete Mage) can explicitly counterspell. Even in these cases, however, they still cannot counterspell other spell-like abilities, only spells.