Synonyms for Exigent (same or very similar meaning)
WordNet sense 1 (demanding attention):
crying, insistent, instant, clamant
WordNet sense 2 (requiring attention or action):
imperative
WordNet sense 3 (requiring precise accuracy):
exacting
WordNet sense 4 (requiring more than usually expected or thought due; especially great patience and effort and skill):
demanding
From the ODE community, based on WordNet
add/edit