Short answer: No, but with a LOT of considerations along the way
Let's break down what is happening:
- Find a beast/plant with no intelligence or an intelligence of 3 or less
- Cast Awaken to make it smart and gain a language
- Cast Feeblemind to make it dumb again
- Cast Awaken again to give it a new language
- Cast Greater Restoration to remove Feeblemind
1. Find a beast/plant with no intelligence or an intelligence of 3 or less
We'll use a black bear as an example... It has an Intelligence of 2 so it qualifies.
2. Cast Awaken to make it smart and gain a language
Per Awaken, after spending 8 hours (casting time) the target:
gains an Intelligence of 10. The target also gains the ability to speak one language you know.
The awakened beast or plant is charmed by you for 30 days or until you or your companions do anything harmful to it.
So now we have a friendly bear that can speak Common.
3. Cast Feeblemind to make it dumb again
Here is one sticking point:
- This is an attack with an average damage of 14; enough to kill some beasts, but the bear has 19 so it would survive all but the best of dice rolls (4d6 maxes at 24). The bear is no longer your friend.
- If they then pass the saving throw, you now have an angry and intelligent bear on your hands.
- If they fail the save...
- The creature's Intelligence and Charisma scores become 1.
- The creature can't cast spells, activate magic items, understand language, or communicate in any intelligible way. The creature can, however, identify its friends, follow them, and even protect them.
Assuming the bear is still alive, and you can calm it down for 8 hours, go on to step 4...
4. Cast Awaken again to give it a new language
And here is the problem, Feeblemind caused TWO conditions; lower ability score AND no way to understand language (or cast spells, etc). You have temporarily lowered the bears Intelligence back below 3 again.
But here is where the tricky logic comes into play.
- Awaken works on plants and beasts even if they have no intelligence, and presumably, no language.
- Awaken gives the target an Intelligence Score of 10.
But it doesn't counteract Feeblemind!
Feeblemind acts like a curse in that it must be removed (either by spell or a distant future saving throw) to be cured, not by masking the symptoms. As pointed out in this question, the headband would give you an Intelligence Score or 19, but you're still the village idiot and cannot understand language or communicate. But you haven't lost any languages you speak.
And as pointed out elsewhere, per the DMG, at the end of the Combat section it specifies what happens when game effects overlap:
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap. For example, if a target is ignited by a fire elemental’s Fire Form trait, the ongoing fire damage doesn’t increase if the burning target is subjected to that trait again. Game features include spells, class features, feats, racial traits, monster abilities, and magic items.
Feeblemind does not cancel out Awaken, only counteracts some of its features. So the first casting is still in effect. Which means that at the most basic level, since Awaken never expires, any future casting would be considered overlapping with the first. Now you're left deciding if the first casting is canceled out completely (resetting Int and losing the language) or that the first casting is "most potent" and therefore cannot be undone 1.
Which brings us back to the point of the question; can the target learn a second (or more) language while under the effects of Feeblemind?
As a DM, I would rule no.
First off, you cannot have two effects of the same name (in this case Awaken) active on the same thing as the same time. So you can't have multiple _Awaken_s on a single creature; whether cast by one caster or multiple casters.
Secondly, as the spell states, "The target also gains the ability to speak one language you know." They know one language the caster speaks, but due to Feeblemind they are unable to understand it. So you could cast Awaken again, but would not bring about the results you hoped for.
1 It may be possible to be undone via Dispel Magic, Wish, and other similar means, but that still resets everything back to the beginning; a normal beast with no increase in Int and no language (beyond its original language if it had one).