Does there exist a (pure) shellable simplicial complex $\Delta$ with the
following property? There is some facet $F$ of $\Delta$ such that no
shelling can begin with $F$.


This condition is easily seen to be equivalent to the existence of a
shellable simplicial complex $\Delta'$ with at least two facets, such
that for some facet $F'$ every shelling of $\Delta'$ ends with $F'$.



Shellable simplicial complex with restriction on shellings
The answer to your question is "yes". Such complexes were ~~first~~ exhibited by Hachimori in his PhD thesis. See the clear explanation on his webpage:  

[http://infoshako.sk.tsukuba.ac.jp/~hachi/math/library/nonextend\_eng.html](http://infoshako.sk.tsukuba.ac.jp/%7Ehachi/math/library/nonextend_eng.html)  

The $f$-vector of this complex is (1,7,19,13). I suspect that 7 vertices is minimal. UPDATE: I later noticed that Simon earlier exhibited an example of similar type in Appendix F of his 1994 paper:  

*Simon, Robert Samuel*, [**Combinatorial properties of “cleanness”**](http://dx.doi.org/10.1006/jabr.1994.1191), J. Algebra 167, No. 2, 361-388 (1994). [ZBL0855.13013](https://zbmath.org/?q=an:0855.13013).  
Hachimori's webpage also has a fairly nice presentation of Simon's example:  

[http://infoshako.sk.tsukuba.ac.jp/~hachi/math/library/simon\_eng.html](http://infoshako.sk.tsukuba.ac.jp/%7Ehachi/math/library/simon_eng.html)


Adiprasito, Benedetti and Lutz have meanwhile (since you first asked the question) extended an idea similar to Hachimori's to arbitrary dimension in their paper:  
 *Adiprasito, Karim A.; Benedetti, Bruno; Lutz, Frank H.*, [**Extremal examples of collapsible complexes and random discrete Morse theory**](http://dx.doi.org/10.1007/s00454-017-9860-4), Discrete Comput. Geom. 57, No. 4, 824-853 (2017). [ZBL1365.05305](https://zbmath.org/?q=an:1365.05305).


Complexes having the property of Hachimori's have arisen recently as a tool. They were used as one of the main building blocks to show that the SHELLABILITY problem is NP-complete in [this paper](https://arxiv.org/abs/1711.08436) of Goaoc, Paták, Patáková, Tancer, and Wagner. (The paper appeared in a conference proceeding, though it doesn't seem to yet be on zbMath.)


I'll mention parenthetically that Goaoc, Paták, Patáková, Tancer, and Wagner require also some other building blocks of the same flavor as Hachimori's complex. Their constructions for these are a bit complicated for my taste, and they don't include all of the details in the above-linked paper. My PhD student Andrés Santamariá Galvis has some simpler constructions that are based on Hachimori's example.


