Premature optimization is the root of all evil (D. Knuth)

So, before we look at how to do this fast, let's look at how to do it correctly :-)

What we are looking for is a word, that is, a sequence of characters between word boundaries, which is followed by one or more instances of itself. So let's build a RE for that :

use strict;

my $test = "alpha beta  beta gamma gamma gamma delta";

# /(\b\w+\b)/ matches a single word :
print "Word in string : $1\n" while $test =~ /(\b\w+\b)/g;

# but we want such a word, followed by (whitespace and then) the word 
+again :
print "Double word : $1\n" if $test =~ /(\b\w+\b)\s*\1/;

# but we not only want to catch duplicates, we also want to catch
# multiple repetitions :
print "Double or more word : $1\n" if $test =~ /(\b\w+\b)(\s*\1)+/;

# And since we're throwing it away anyway, there's no need to actually
# capture stuff into $2 :
print "Double or more word : $1\n" if $test =~ /(\b\w+\b)(?:\s*\1)+/;

# Now, here's the whole RE to seek out repeating words
# and collapse them into one word.
$test =~ s/(\b\w+\b)(?:\s*\1)+/$1/g;

print $test;
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In this short time, I didn't yet give much thought about optimization, and I think that a regular expression string replace might possibly not be the fastest solution, some string scanning might be faster, but as you don't give much context, I won't be of much help in that departement.

Update: Changed attribution from Kernighan to Knuth