Relation between fully-extended TQFT and a "topless" TQFT 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.
 A: 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 that the theory can be extended to a full "3-2-1-0" theory.
A: With Douglas and Schommer-Pries construct such a 210 TFT for every finite tensor category (in the sense of Etingof-Ostrik).  When the category is not semisimple there's no 3210 TFT.
My understanding from conversations with Kevin Walker is that one should have a similar story for (some) infinite tensor categories, but in our setup we are not yet able to prove such a result rigorously (Kevin's formalization of local TFT is somewhat different).
