How to make stacked circle plot without coord_polar

If you want simple radius bands, perhaps something like this would work as you pictured it in your question:

# your original sample data
x <- 100 - abs(rnorm(1e6, 0, 5))
y <- 50 + rnorm(1e6, 0, 3)
dist <- sqrt((x - 100)^2 + (y - 50)^2)

nbr_bands <- 6  # set nbr of bands to plot 

# calculate width of bands
band_width <- max(dist)/(nbr_bands-1)

# dist div band_width yields an integer 0 to nbr bands
# as.factor makes it categorical, which is what you want for the plot
band = as.factor(dist %/% (band_width))

library(dplyr)
library(ggplot2)
data.frame(x, y, band) %>%  
  ggplot() + geom_point(aes(x, y, color = band)) + coord_fixed() +
  theme_dark()  # dark theme

Edit to elaborate:

As you first attempted, it would be nice to use the very handy cut() function to calculate the radius color categories.

One way to get categorical (discrete) colors, rather than continuous shading, for your plot color groups is to set your aes color= to a factor column.

To directly get a factor from cut() you may use option ordered_result=TRUE:

band <- cut(dist, nbr_bands, ordered_result=TRUE, labels=1:nbr_bands)  # also use `labels=` to specify your own labels

data.frame(x, y, band) %>%
  ggplot() + geom_point(aes(x, y, color = band)) + coord_fixed() 

Or more simply you may use cut() without options and convert to a factor using as.factor():

band <- as.factor( cut(dist, nbr_bands, labels=FALSE) )

data.frame(x, y, band) %>%
  ggplot() + geom_point(aes(x, y, color = band)) + coord_fixed() 

Sounds like you may find the circle & arc plotting functions from the ggforce package useful:

# data
set.seed(1234)
df <- data.frame(x = 100 - abs(rnorm(1e6, 0, 5)),
                 y = 50 + rnorm(1e6, 0, 3)) %>%  
  mutate(dist = sqrt((x - 100)^2 + (y - 50)^2)) %>%
  mutate(z = exp(-(dist / 8)^2))

# define cut-off values
cutoff.values <- seq(0, ceiling(max(df$dist)), by = 5)

df %>%
  # calculate the mean z for each distance band
  mutate(dist_fct = cut(dist, cutoff.values)) %>%
  group_by(dist_fct) %>%
  summarise(z = mean(z)) %>%
  ungroup() %>%

  # add the cutoff values to the dataframe for inner & outer radius
  arrange(dist_fct) %>%
  mutate(r0 = cutoff.values[-length(cutoff.values)],
         r = cutoff.values[-1]) %>%

  # add coordinates for circle centre
  mutate(x = 100, y = 50) %>%

  # plot
  ggplot(aes(x0 = x, y0 = y, 
             r0 = r0, r = r, 
             fill = z)) +
  geom_arc_bar(aes(start = 0, end = 2 * pi), 
               color = NA) + # hide outline

  # force equal aspect ratio in order to get true circle
  coord_equal(xlim = c(70, 100), expand = FALSE)

Plot generation took <1s on my machine. Yours may differ.

I'm not sure this satisfies everything, but it should be a start. To cut down on the time for plotting, I'm summarizing the data into a grid, which lets you use geom_raster. I don't entirely understand the breaks and everything you're using, so you might want to tweak some of how I divided the data for making the distinct bands. I tried out a couple ways with cut_interval and cut_width--this would be a good place to plug in different options, such as the number or width of bands.

Since you mentioned getting the average z for each band, I'm grouping by the gridded x and y and the cut dist, then using mean of z for setting bands. I threw in a step to make labels like in the example--you probably want to reverse them or adjust their positioning--but that comes from getting the number of each band's factor level.

library(tidyverse)
set.seed(555)
n <- 1e6

df <- data_frame(
  x = 100 - abs(rnorm(n, 0, 5)), 
  y = 50 + rnorm(n, 0, 3), 
  dist = sqrt((x - 100)^2 + (y - 50)^2), 
  z = exp(-(dist / 8)^2)
) %>%
  mutate(brk = cut(dist, seq(0, max(dist), by = 5), include.lowest = T))

summarized <- df %>%
  filter(!is.na(brk)) %>%
  mutate(x_grid = floor(x), y_grid = floor(y)) %>%
  group_by(x_grid, y_grid, brk) %>%
  summarise(avg_z = mean(z)) %>%
  ungroup() %>%
  # mutate(z_brk = cut_width(avg_z, width = 0.15)) %>%
  mutate(z_brk = cut_interval(avg_z, n = 9)) %>%
  mutate(brk_num = as.numeric(z_brk))

head(summarized)
#> # A tibble: 6 x 6
#>   x_grid y_grid brk         avg_z z_brk           brk_num
#>    <dbl>  <dbl> <fct>       <dbl> <fct>             <dbl>
#> 1     75     46 (20,25] 0.0000697 [6.97e-05,0.11]       1
#> 2     75     47 (20,25] 0.000101  [6.97e-05,0.11]       1
#> 3     75     49 (20,25] 0.0000926 [6.97e-05,0.11]       1
#> 4     75     50 (20,25] 0.0000858 [6.97e-05,0.11]       1
#> 5     75     52 (20,25] 0.0000800 [6.97e-05,0.11]       1
#> 6     76     51 (20,25] 0.000209  [6.97e-05,0.11]       1

To make the labels, summarize that data to have a single row per band--I did this by taking the minimum of the gridded x, then using the average of y so they'll show up in the middle of the plot.

labels <- summarized %>%
  group_by(brk_num) %>%
  summarise(min_x = min(x_grid)) %>%
  ungroup() %>%
  mutate(y_grid = mean(summarized$y_grid))

head(labels)
#> # A tibble: 6 x 3
#>   brk_num min_x y_grid
#>     <dbl> <dbl>  <dbl>
#> 1       1    75   49.7
#> 2       2    88   49.7
#> 3       3    90   49.7
#> 4       4    92   49.7
#> 5       5    93   49.7
#> 6       6    94   49.7

geom_raster is great for these situations where you have data in an evenly spaced grid that just needs uniform tiles at each position. At this point, the summarized data has 595 rows, instead of the original 1 million, so the time to plot shouldn't be an issue.

ggplot(summarized) +
  geom_raster(aes(x = x_grid, y = y_grid, fill = z_brk)) +
  geom_label(aes(x = min_x, y = y_grid, label = brk_num), data = labels, size = 3, hjust = 0.5) +
  theme_void() +
  theme(legend.position = "none", panel.background = element_rect(fill = "gray40")) +
  coord_fixed() +
  scale_fill_brewer(palette = "PuBu")

^{Created on 2018-11-04 by the reprex package (v0.2.1)}