In base R, you can just use ave
to create a "count" column. You don't really specify how you want to deal with "Col3", so I've left that out:
mydf$count <- ave(rep(1, nrow(mydf)),
mydf[c("Col1", "Col2", "Day")],
FUN = length)
mydf
# Col1 Col2 Col3 Day count
# 1 12 2 1.40311e+16 70 3
# 2 10 3 1.40311e+16 6 2
# 3 7 4 1.40311e+16 11 1
# 4 12 2 1.40311e+16 70 3
# 5 12 2 1.40311e+16 70 3
# 6 10 3 1.40311e+16 6 2
unique(mydf[c("Col1", "Col2", "Day", "count")])
# Col1 Col2 Day count
# 1 12 2 70 3
# 2 10 3 6 2
# 3 7 4 11 1