What role does the notion of subsidiarity place in the Church, given it’s rather famous for its hierarchy (“great chain of being”).
That should be a “?” not a “.”. 
I suspect, based on 1,950 years of accrued interpretation of roles, that one could make a good case that subsidiarity plays a strong role in the church or that it is irrelevant to it.
On the pro side, based on the notion of “most immediate level of competence,” we see that many church actions (and decisions) are carried out at that point. Only the ordinary (the bishop in charge of the diocese) has the authority to perform the sacrament of Confirmation, yet they routinely confer upon their assistants that same “power” and it is a common practice to confer upon the diocesan priest the authority to Confirm candidates at the Rite of Initiation held at the Easter Vigil.
Against that position, the “level of competence” for sacraments such as Reconciliation/Penance and the Eucharist has been set at the priesthood and no priest, bishop, or pope can grant a deacon (much less a layperson) the authority to preside at those sacraments. That level of competence has been set through the last couple of millennia. Do they recognize some genuine level of competence inherent in the office? Or are they, in some fashion, arbitrary?
If we step away from the sacraments, I think we see a clear break with subsidiarity, although they are not capricious breaks and, depending on one’s perspective, one might still argue that subsidiarity is still occurring.
For example, in the U.S., all financial authority is vested in the diocese which is incorporated under the various state laws. Individual parishes cannot choose to purchase land or take out loans to build or expand buildings and are not subject to individual lawsuits (which must be filed against the diocese). It might be argued that this is a failure of subsidiarity. What prevents a parish from having the competence to decide when and how to expand or to be held liable if it fails to maintain a safe structure or pay its bills?
Of course, the lack of authority is the direct result of individual parishes, through the years, taking on projects they could not handle or parishoners trying to hold their buildings “hostage” when choosing to defy the bishop on various occasions for various reasons. There is also the conflict that a pastor may choose to operate the parish in defiance of his direct boss–spiritual and legal–the bishop, so that vesting all financial authority in the bishop simply reduces the potential for internal issues boiling over into public scandal.
If you can flesh out your question, we might get to an answer closing to what you hoped to learn.
To nitpick, in New York Roman Catholic parishes may be incorporated under article 5 of the Religious Corporations Law (other denominations have articles particular to their heirarchical structure). The board of trustees of each such parish consists of the Bishop (or Archbishop), the vicar general of the diocese, the rector of the parish and two laymen selected by the above, though all acts must be approved by the bishop/archbishop (or the vicar general, if the bishop is unavailable). Incorporated parishes may sue and be sued, purchase land, take out loans, and generally have the additional powers of other types of corporations in the state.