Consider 3-dimensional TQFTs for example. One version of them is the 
3-2-1-0 fully extended TQFT. **Do we have another version: 2-1-0 extended "TQFT"?** 


If yes, **do we have an example of 2-1-0 extended TQFT that is not
3-2-1-0 fully extended TQFT?**


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By 2-1-0 extended 3-dim "TQFT", we mean that
we can assign a Hilbert space to every closed orientable 2-manifold, but we do not require the path integral to be well defined for every closed orientable 3-manifold. Certainly, there is a higher dimensional analogue of this.


Unitary condition: The Hilbert space mentioned above has a well define inner product so that the norm are all positive. Also the Hilbert space has a finite dimension on all closed surfaces.



Relation between fully-extended TQFT and a "topless" TQFT
If I understand you correctly, your "2-1-0" TQFTs are what are frequently called "2+$\epsilon$-dimensional TQFTs" in the mathematical literature. (The $\epsilon$ means that very thin 3-manifolds, e.g. the mapping cylinder of a homeomorphism of 2-manifolds, can have their path integral defined.)


If you try to construct a Turaev-Viro TQFT (Levin-Wen model) based on $Rep(U\_q(g))$ for q **not** a root of unity (so there are infinitely many simple objects), then you can construct an (infinite-dimensional) Hilbert space for any 2-manifold, and also assign a 1-category to 1-manifolds and 2-category to 0-manifolds. But you cannot construct the path integral of, say, $Y\times S^1$, since this should be equal to the dimension of $Z(Y)$ which is infinite (unless $Y$ is very simple). So this is an example of a "2-1-0" theory which is not a "3-2-1-0" theory.




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In response to your comment asking for a unitary example with finite-dimensional Hilbert spaces:


If your definition of "unitary" in this context is the same as mine, then the answer is that any finite unitary 2+$\epsilon$ ("x-2-1-0") theory extends to a full "3-2-1-0" theory. More specifically, if the input 2-category (e.g. tensor category) has a collection of positive-definite inner products which are compatible with the tensor category structure (I assume "unitary" implies this) and the Hilbert space for any surface is finite-dimensional, then it follows from Theorem 6.3.1 of [this](http://canyon23.net/math/tc.pdf) that the theory can be extended to a full "3-2-1-0" theory.


