Comment. A surprisingly broad question. 

[This page](http://www.statistics4u.com/fundstat_eng/cc_corr_coeff_distri.html)
shows simulation results for correlations of two
independent standard uniform distributions. Results seem to differ
greatly for various (especially small) numbers $n$ of pairs of points.
Each histogram represents $100,000$ values of $r.$

[![enter image description here][1]][1]

For normal data, of course, there is Fisher's work on the distribution
of $r.$ (See [Wikipedia](https://en.wikipedia.org/wiki/Pearson_correlation_coefficient).)

When you say 'average correlation', do you mean $E(r)$ or $E(|r|)?$
When you say 'maximum correlation', what sample sizes do you have in mind to keep the maximum below 1? To start: I hope you can specify a few cases
of greatest interest (dimensions, sample sizes, uniform vs. normal vs. what other distributions?).

Addendum per @whuber's Comment: Further experimentation. Five observations on each of five independent uniform variables.

    set.seed(906)
    n = 5;  m = 5;  u = runif(n*m)
    DTA = matrix(u, nrow=m)
    cor(DTA)
               [,1]       [,2]       [,3]        [,4]        [,5]
    [1,]  1.0000000  0.1957409 -0.8233160 -0.39093048 -0.62212314
    [2,]  0.1957409  1.0000000 -0.1255770  0.62306226 -0.61205725
    [3,] -0.8233160 -0.1255770  1.0000000  0.36625813  0.22424637
    [4,] -0.3909305  0.6230623  0.3662581  1.00000000  0.08519501
    [5,] -0.6221231 -0.6120572  0.2242464  0.08519501  1.00000000
    max(abs(cor(DTA)-diag(5)))
    [1] 0.823316

Highest absolute correlation happens to be about $|r| \approx 0.82,$ between
variables 1 and 3. Matrix plot shows all ${5 \choose 2}$ pairs---see center top row. (Max absolute correlations this high are not rare. Tried several runs before
this one with `set.seed` to post.)

    pairs(DTA)

[![enter image description here][2]][2]


  [1]: https://i.sstatic.net/OEYh6.png
  [2]: https://i.sstatic.net/lC9Xp.png