Directlabels package-- labels do not fit in plot area

In my opinion, direct labels is the way to go. Indeed, I would position labels at the beginning and at the end of the lines, creating space for the labels using expand(). Also note that with the labels, there is no need for the legend.

This is similar to answers here and here.

library(ggplot2)
library(directlabels)
library(grid)
library(tidyr)

x <- seq(1:100)
y <- cumsum(rnorm(n = 100, mean = 6, sd = 15))
y2 <- cumsum(rnorm(n = 100, mean = 2, sd = 4))
data <- as.data.frame(cbind(x, y, y2))
names(data) <- c("month", "stocks", "bonds")
tidy_data <- gather(data, month)
names(tidy_data) <- c("month", "asset", "value")

ggplot(tidy_data, aes(x = month, y = value, colour = asset, group = asset)) + 
     geom_line() + 
     scale_colour_discrete(guide = 'none')  + 
     scale_x_continuous(expand = c(0.15, 0)) +
     geom_dl(aes(label = asset), method = list(dl.trans(x = x + .3), "last.bumpup")) +
     geom_dl(aes(label = asset), method = list(dl.trans(x = x - .3), "first.bumpup")) + 
     theme_bw() 

If you prefer to push the labels into the plot margin, direct labels will do that. But because the labels are positioned outside the plot panel, clipping needs to be turned off.

p1 <- ggplot(tidy_data, aes(x = month, y = value, colour = asset, group = asset)) + 
     geom_line() + 
     scale_colour_discrete(guide = 'none')  + 
     scale_x_continuous(expand = c(0, 0)) +
     geom_dl(aes(label = asset), method = list(dl.trans(x = x + .3), "last.bumpup")) +
     theme_bw() +
     theme(plot.margin = unit(c(1,4,1,1), "lines")) 

# Code to turn off clipping
gt1 <- ggplotGrob(p1)
gt1$layout$clip[gt1$layout$name == "panel"] <- "off"
grid.draw(gt1)

This effect can also be achieved using geom_text (and probably also annotate), that is, without the need for direct labels.

p2 = ggplot(tidy_data, aes(x = month, y = value, group = asset, colour = asset)) +
  geom_line() + 
  geom_text(data = subset(tidy_data, month == 100), 
      aes(label = asset, colour = asset, x = Inf, y = value), hjust = -.2) +
  scale_x_continuous(expand = c(0, 0)) +
  scale_colour_discrete(guide = 'none')  +  
  theme_bw() +  
  theme(plot.margin = unit(c(1,3,1,1), "lines"))  

# Code to turn off clipping
gt2 <- ggplotGrob(p2)
gt2$layout$clip[gt2$layout$name == "panel"] <- "off"
grid.draw(gt2)

Since you didn't provide a reproducible example, it's hard to say what the best solution is. However, I would suggest trying to manually adjust the x-scale. Use a "buffer" increase the plot area.

#generate data frame with random data, for illustration and plot:
p <- ggplot(tidy_data, aes(x = month, y = value, colour = asset)) + 
geom_line() + 
geom_dl(aes(colour = asset, label = asset), method = "last.points") + 
theme_bw() +
xlim(minimum_value, maximum_value + buffer)

Using scale_x_discrete() or scale_x_continuous() would likely also work well here if you want to use the direct labels package. Alternatively, annotate or a simple geom_text would also work well.