If we need the "Count" and "Names" corresponding to the max "Count", we loop through the rows of the dataset (using apply with MARGIN = 1), use table to get the frequency, extract the maximum value from it and the names corresponding to the maximum value, rbind it and cbind with the original dataset.
cbind(df1, do.call(rbind, apply(df1[-1], 1, function(x) {
x1 <- table(x)
data.frame(Count = max(x1), Names=names(x1)[which.max(x1)])})))
# Name Mon Tue Wed Thu Fri Sat Sun Count Names
#1 John Apple Orange Apple Banana Apple Apple Orange 4 Apple
#2 Ricky Banana Apple Banana Banana Banana Banana Apple 5 Banana
#3 Alex Apple Orange Orange Apple Apple Orange Orange 4 Orange
#4 Robbin Apple Apple Apple Apple Apple Banana Banana 5 Apple
#5 Sunny Banana Banana Apple Apple Apple Banana Banana 4 Banana
Or we can use data.table
library(data.table)
setDT(df1)[, c("Names", "Count") := {tbl <- table(unlist(.SD))
.(names(tbl)[which.max(tbl)], max(tbl))}, by = Name]