library(dplyr)

df <- data.frame(grp_A = c(13, NA, NA, NA, NA, 20, NA),
             grp_B = c(NA, 59, 66, NA, NA, NA, NA),
             grp_C = c(NA, NA, NA, 23, 42, NA, NA))

df$value <- apply(select(df, grp_A, grp_B, grp_C), 1,
              function(x) x[!is.na(x)])

df$value <- df[cbind(1:nrow(df), max.col(!is.na(df)))]

df
#  grp_A grp_B grp_C value
#1    13    NA    NA    13
#2    NA    59    NA    59
#3    NA    66    NA    66
#4    NA    NA    23    23
#5    NA    NA    42    42
#6    20    NA    NA    20
#7    NA    NA    NA    NA

max.col(!is.na(df))
#[1] 1 2 2 3 3 1 2

library(dplyr)

df <- data.frame(grp_A = c(13, NA, NA, NA, NA, 20, NA),
                 grp_B = c(NA, 59, 66, NA, NA, NA, NA),
                 grp_C = c(NA, NA, NA, 23, 42, NA, NA))

mutate(df, value = Reduce(coalesce, df))

grp_A grp_B grp_C value
1    13    NA    NA    13
2    NA    59    NA    59
3    NA    66    NA    66
4    NA    NA    23    23
5    NA    NA    42    42
6    20    NA    NA    20
7    NA    NA    NA    NA

df$value <- rowSums(df, na.rm = T)

df[df$value == 0, ] <- NA