The Collaborative International Dictionary of English v.0.48:

Besiege \Be*siege"\, v. t. [imp. & p. p. Besieged; p. pr. &
   vb. n. Besieging.] [OE. bisegen; pref. be- + segen to
   siege. See Siege.]
   To beset or surround with armed forces, for the purpose of
   compelling to surrender; to lay siege to; to beleaguer; to
   beset.
   [1913 Webster]

         Till Paris was besieged, famished, and lost. --Shak.
   [1913 Webster]

   Syn: To environ; hem in; invest; encompass.
        [1913 Webster]

WordNet (r) 3.0 (2006):

besiege
    v 1: surround so as to force to give up; "The Turks besieged
         Vienna" [syn: besiege, beleaguer, surround, hem in,
         circumvent]
    2: cause to feel distressed or worried; "She was besieged by so
       many problems that she got discouraged"
    3: harass, as with questions or requests; "The press
       photographers besieged the movie star"

Moby Thesaurus II by Grady Ward, 1.0:

86 Moby Thesaurus words for "besiege":
   apply pressure, assail, assault, attack, beleaguer, beset,
   blandish, block, block off, block up, blockade, bound, box in, bug,
   buttonhole, cage, cajole, chamber, close in, coax, compass,
   contain, coop, coop in, coop up, cordon, cordon off, corral,
   cut off, dun, encircle, enclose, encompass, enshrine, envelop,
   exert pressure, fence in, harass, harry, hedge in, hem in,
   house in, importune, impound, imprison, incarcerate, include,
   inundate, invest, jail, kennel, lay siege to, leaguer, mew, mew up,
   nag, nag at, overwhelm, pen, pen in, pester, petition, plague, ply,
   pocket, press, pressure, push, quarantine, rail in, shrine,
   shut in, shut up, soften up, stable, sue, surround, tease, trap,
   urge, wall in, wheedle, work on, wrap, yard, yard up