The Collaborative International Dictionary of English v.0.48:

Beseech \Be*seech"\, n.
   Solicitation; supplication. [Obs. or Poetic] --Shak.
   [1913 Webster]

The Collaborative International Dictionary of English v.0.48:

Beseech \Be*seech"\, v. t. [imp. & p. p. Besought; p. pr. &
   vb. n. Beseeching.] [OE. bisechen, biseken (akin to G.
   besuchen to visit); pref. be- + sechen, seken, to seek. See
   Seek.]
   1. To ask or entreat with urgency; to supplicate; to implore.
      [1913 Webster]

            I beseech you, punish me not with your hard
            thoughts.                             --Shak.
      [1913 Webster]

            But Eve . . . besought his peace.     --Milton.
      [1913 Webster]

   Syn: To beg; to crave.

   Usage: To Beseech, Entreat, Solicit, Implore,
          Supplicate. These words agree in marking that sense
          of want which leads men to beg some favor. To solicit
          is to make a request, with some degree of earnestness
          and repetition, of one whom we address as a superior.
          To entreat implies greater urgency, usually enforced
          by adducing reasons or arguments. To beseech is still
          stronger, and belongs rather to the language of poetry
          and imagination. To implore denotes increased fervor
          of entreaty, as addressed either to equals or
          superiors. To supplicate expresses the extreme of
          entreaty, and usually implies a state of deep
          humiliation. Thus, a captive supplicates a conqueror
          to spare his life. Men solicit by virtue of their
          interest with another; they entreat in the use of
          reasoning and strong representations; they beseech
          with importunate earnestness; they implore from a
          sense of overwhelming distress; they supplicate with a
          feeling of the most absolute inferiority and
          dependence.
          [1913 Webster]

Moby Thesaurus II by Grady Ward, 1.0:

36 Moby Thesaurus words for "beseech":
   adjure, appeal, appeal to, beg, call for help, call on, call upon,
   clamor for, commune with God, conjure, crave, cry for, cry on,
   cry to, entreat, give thanks, impetrate, implore, importune,
   imprecate, invoke, kneel to, make supplication, obtest,
   offer a prayer, petition, plead, plead for, plead with, pray,
   pray over, recite the rosary, return thanks, run to, say grace,
   supplicate