Is there a R function that applies a function to each pair of columns?

92% of the time is being spent in cor.test.default and routines it calls so its hopeless trying to get faster results by simply rewriting Papply (other than the savings from computing only those above or below the diagonal assuming that your function is symmetric in x and y).

> M <- matrix(rnorm(100*300),300,100)
> Rprof(); junk <- Papply(M,function(x,y) cor.test( x, y)$p.value); Rprof(NULL)
> summaryRprof()
$by.self
                 self.time self.pct total.time total.pct
cor.test.default      4.36    29.54      13.56     91.87
# ... snip ...

It wouldn't be faster, but you can use outer to simplify the code. It does require a vectorized function, so here I've used Vectorize to make a vectorized version of the function to get the correlation between two columns.

df <- data.frame(x=rnorm(100),y=rnorm(100),z=rnorm(100))
n <- ncol(df)

corpij <- function(i,j,data) {cor.test(data[,i],data[,j])$p.value}
corp <- Vectorize(corpij, vectorize.args=list("i","j"))
outer(1:n,1:n,corp,data=df)

I'm not sure if this addresses your problem in a proper manner, but take a look at William Revelle's psych package. corr.test returns list of matrices with correlation coefs, # of obs, t-test statistic, and p-value. I know I use it all the time (and AFAICS you're also a psychologist, so it may suite your needs as well). Writing loops is not the most elegant way of doing this.

> library(psych)
> ( k <- corr.test(mtcars[1:5]) )
Call:corr.test(x = mtcars[1:5])
Correlation matrix 
       mpg   cyl  disp    hp  drat
mpg   1.00 -0.85 -0.85 -0.78  0.68
cyl  -0.85  1.00  0.90  0.83 -0.70
disp -0.85  0.90  1.00  0.79 -0.71
hp   -0.78  0.83  0.79  1.00 -0.45
drat  0.68 -0.70 -0.71 -0.45  1.00
Sample Size 
     mpg cyl disp hp drat
mpg   32  32   32 32   32
cyl   32  32   32 32   32
disp  32  32   32 32   32
hp    32  32   32 32   32
drat  32  32   32 32   32
Probability value 
     mpg cyl disp   hp drat
mpg    0   0    0 0.00 0.00
cyl    0   0    0 0.00 0.00
disp   0   0    0 0.00 0.00
hp     0   0    0 0.00 0.01
drat   0   0    0 0.01 0.00

> str(k)
List of 5
 $ r   : num [1:5, 1:5] 1 -0.852 -0.848 -0.776 0.681 ...
  ..- attr(*, "dimnames")=List of 2
  .. ..$ : chr [1:5] "mpg" "cyl" "disp" "hp" ...
  .. ..$ : chr [1:5] "mpg" "cyl" "disp" "hp" ...
 $ n   : num [1:5, 1:5] 32 32 32 32 32 32 32 32 32 32 ...
  ..- attr(*, "dimnames")=List of 2
  .. ..$ : chr [1:5] "mpg" "cyl" "disp" "hp" ...
  .. ..$ : chr [1:5] "mpg" "cyl" "disp" "hp" ...
 $ t   : num [1:5, 1:5] Inf -8.92 -8.75 -6.74 5.1 ...
  ..- attr(*, "dimnames")=List of 2
  .. ..$ : chr [1:5] "mpg" "cyl" "disp" "hp" ...
  .. ..$ : chr [1:5] "mpg" "cyl" "disp" "hp" ...
 $ p   : num [1:5, 1:5] 0.00 6.11e-10 9.38e-10 1.79e-07 1.78e-05 ...
  ..- attr(*, "dimnames")=List of 2
  .. ..$ : chr [1:5] "mpg" "cyl" "disp" "hp" ...
  .. ..$ : chr [1:5] "mpg" "cyl" "disp" "hp" ...
 $ Call: language corr.test(x = mtcars[1:5])
 - attr(*, "class")= chr [1:2] "psych" "corr.test"