Using data.table package, here is a way to obtain what you want:
require(data.table)
setDT(dt)[, Seq:=1:.N, by=IDPath]
# or, as mentioned by @DavidArenburg
setDT(dt)[, Seq:=seq_len(.N), by=IDPath]
dt
# IDPath LogTime Seq
#1: AADS 19-06-2015 01:57 1
#2: AADS 19-06-2015 01:55 2
#3: AADS 19-06-2015 01:54 3
#4: AADS 19-06-2015 01:53 4
#5: DHSD 19-06-2015 12:57 1
#6: DHSD 19-06-2015 10:58 2
#7: DHSD 19-06-2015 09:08 3
#8: DHSD 19-06-2015 08:41 4
