Consider the following:
I’ll probably stay at home.
I’ll probably stay home.
Is the second sentence still grammatically correct? Is there any difference at all?
Both are correct. There are instances where they mean the same thing and some instances where they don’t.
If you were telling someone that you didn’t intend to go out tonight, you could use either.
“Do you want to go with us to a restaurant tonight?”
“No, I think I’ll stay home.”
“No, I think I’ll stay at home.”
However, if someone were asking where you were staying, where the answer might be “a hotel” or “a friend’s house”, you would definitely say “at home”.
“Are you going to get a hotel room for the conference?”
“No, I’m going to stay at home.”
“No, I’m going to stay home.”
(Note that if you did say the struck-through version, that would tend to imply that you weren’t going to go to the conference at all.)
However, that “home” is uncommon in that it also functions as an adverb. You can never leave the preposition out with other similar nouns. For example, this is clearly wrong:
“I’ll probably stay work.”
In that case, you have to create an adverbial prepositional phrase:
“I’ll probably stay at work.”