The most common and most likely answer to this crossword clue is the 3 letter word BUT
I've seen this crossword clue in The Times
B
U
T
Buddy explains!

The straight part
We know that "I object" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case i object = but.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does BUT mean?
BUT noun- An instance or example of using the word "but".
- The outer room of a small two-room cottage.
- A limit; a boundary.
- The end; especially the larger or thicker end, or the blunt, in distinction from the sharp, end; the butt.
- Use the word "but".
- Merely, only, just.
- Though, however.
- Used as an intensifier.
- Apart from, except (for), excluding.
- (obsolete outside Scotland) Outside of.
- On the contrary, rather (as a regular adversative conjunction, introducing a word or clause in contrast or contradiction with the preceding negative clause or sentence).
- However, although, nevertheless, on the other hand (introducing a clause contrary to prior belief or in contrast with the preceding clause or sentence).
- Except that (introducing a subordinate clause which qualifies a negative statement); also, with omission of the subject of the subordinate clause, acting as a negative relative, "except one that", "except such that".
- Without its also being the case that; unless that (introducing a necessary concomitant).
- Except with; unless with; without.
- Only; solely; merely.
- Until.
- (following a negated expression of improbability) That.