grep1-3  Exercise 2: Repeated Words and Doubled Letters
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cat > text.txt <<'EOF'
This is is a test of the the system
Nothing repeated here
Bookkeeper keeps books
It was the best of times, it was the worst of times
a a a
EOF

echo "Repeated words (any case):"
grep -oiE '\b([a-z]+) \1\b' text.txt
echo
echo "Words with a doubled letter:"
grep -oiE '\b[a-z]*([a-z])\1[a-z]*\b' text.txt

Output:

Repeated words (any case):
is is
the the
a a

Words with a doubled letter:
Bookkeeper
keeps
books

WHY THIS WORKS AS AN ANSWER
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([a-z]+) captures a word and \1 demands the same text again, with -i making the comparison case-insensitive. \b at both ends stops a match from starting or ending inside a word (without the closing \b, 'the then' would wrongly report 'the the': I tried it).
The second command finds a word with a doubled letter: [a-z]* any letters, then ([a-z]) one letter captured, then \1 the same letter, then more letters. \b at both ends makes -o print the whole word rather than just the middle.
'a a a' gives one match, 'a a': matching consumes text, so the third 'a' has nothing left to repeat. Back-references with -o are the way to see exactly what was found.