It depends on what you mean by "know," what you mean by "something," and what you mean by "assumptions." Actually, it depends on a whole lot more than that, but those are the really fun words.
What makes this topic tricky is that the concept of "knowing" is so deeply buried in our language(English) that it's hard to even capture something that describes it acceptably.
The most general answer must be "maybe," but if we restrict ourselves to the most common senses of the words in philosophy, the answer is a resounding "no." In order to find a "maybe" answer we have to step away from the typical definitions.
For example, what is "something" you can know, anyways? Your words imply that being able to make a "statement" is part of the puzzle. So how about we start with a particularly obnoxious statement:
Oh freddled gruntbuggly,
Thy micturations are to me
As plurdled gabbleblotchits on a lurgid bee.
Is that something I can "know?" What does it mean, anyway? The most common answer is that any statement, stated in a language, must be interpreted in some way to arrive at some semantic truth -- something knowable. I must have an understanding of the language.
And the assumption that my interpretation of this statement is correct is an assumption. We can't just handwave it away and say "oh, assume our interpretation of the language is right." If we do that, then we immediately find that we can sneak assumptions into the grammar of the language to hide them from our counting. We didn't really decrease the assumptions, we just moved them.
Also, how do we count them? If I use the language of propositional logic for a moment, if I assume one statement, A∧B∧C, did I actually assume anything less than if I assumed three statements, A, B, and C? Even counting assumptions is a tricky beast. In computing, there's a concept called Kolmogorov complexity which studies how many bits of information it takes to convey something in a particular language. Even then, it's used mostly to prove the impossibility of stating certain things:
In particular, for almost all objects, it is not possible to compute even a lower bound for its Kolmogorov complexity (Chaitin 1964), let alone its exact value.
One fascinating path people have taken is to consider trying to create self-hoisting languages, which can prove their own consistency. This was a fascinating effort in the early and mid 1900's, but what we found was that it is generally not possible. Propositional logic is too weak to be able to admit the self-referential structures required for a language to talk about itself. First Order Logic has to deal with Godel's Incompleteness Theorem, which is notorious. Second Order Logic can indeed talk about itself, but it can't admit proofs of its own correctness. So of our "standard" languages, none of them admit statements without assumptions.
So can you know something without an assumption? Well... maybe. We can show that entire vast swaths of what we'd like to say knowledge "is" cannot operate without an assumption. However, none of the "typical" structures can prove that we're using the right definition of "know" or "assumption" or anything, really. So maybe the definition of "know something without assumptions" that you are using indeed admits such a thing. Or maybe it doesn't. We can't devise the language to prove one way or the other without making an assumption.
I'd like to close with two great resources. One is a beautiful speech by Jon Steele on How to Grow a Language. It's a massively long video, so the transcript may be more palatable. He constructs a language, from the ground up, using a very particular set of rules. I find what he was doing is much in the vein of what you are thinking about.
The second is one of my favorite quotes from Stranger in a Strange Land, by Heinlein. Mike Smith is the subject of this quote, and he was raised Mars, speaking Martian. Only upon coming to Earth did he have to learn English:
Short human words were never like a short Martian word — such as "grok" which forever meant exactly the same thing. Short human words were like trying to lift water with a knife.
And [God] had been a very short word.
