beekeeper
effervescence
...and about 211 others like electioneer, largeheartednesses (and many others with the -nesses ending), reexperience(ed, ing), reengineer(ed, ing), defervescence(s) and others. Try a word generator site. If asterisk represents a wild card then *e*e*e*e*e* would bring up words with five e's and 0 to n other letters in between the e's.
elite, extra, emote
Ensue.
issue
eight
eighteens
nappy, paper
Some examples of a five-letter word ending with 'e' with the second letter of 'u' is"pulsebulgecurvepurge
evokeevadeeludeexileeerieeagleeliteelateemceeetudeeyrieeraseexudeemoteensueelopeentre
epees
earn
east
egis
Taken