度目 is read as どめ (dome) and means the Nth time.
どめ
dome
こくさいおんどめもり
International Temperature Scale
おんどめもり
temperature scale
たびたび
often
たびに
each time
ど
damn
たび
time (three times, each time, etc.)
たびかさなる
to repeat
めいもく
(under the) pretext (of)
ひとめ
(public) notice
め
(an) appearance
ひんもく
item
ちゅうもく
notice