Adding bigger and bigger offsets will bring the values closer together in the absolute scale, but if you similarly scale your axis then counterintuitively the lower values will be bunched together more the larger you make the offset.

x <- c(100, 150, 200, 250, 1500)

par(mfrow=c(1,3), mar=rep(2, 4))
lapply(1:3, \(e) plot(log(x+10^e), x, ann=FALSE))

The solution isn't so much the offset -- the smaller the better but that'll only get you so far. What you want is a log-log(-log? -log?) transform:

## Stack those logs! (more works too)
my_transform <- function(x) log(log(x + 1E-3))
my_inverse <- function(x) exp(exp(x)) - 1E-3

my_trans <-  scales::trans_new("yes_mate", transform = my_transform, inverse = my_inverse)

ggplot(mtcars, aes(x = disp, y = 1, color = disp)) +
  geom_point(size = 5) +
  scale_x_continuous(
    trans = my_trans,
    limits=c(100, 1500),
    breaks = c(100, 500, 1000, 1500)
  )