Questioner:

Please can you explain what is meant by the last line in this quote from ‘Who am I?’ by Sri Ramana Maharshi?

‘By a steady and continuous investigation into the nature of the mind, the mind is transformed into That to which the ‘I’ refers; and that is in fact the Self.

Mind has necessarily to depend for its existence on something gross; it never subsists by itself. It is this mind that is otherwise called the subtle body, ego, jiva or soul.

That which arises in the physical body as ‘I’ is the mind. If one enquires whence the ‘I’ thought in the body arises in the first instance, it will be found that it is from hrdayam or the Heart. That is the source and stay of the mind.

Or again, even if one merely continuously repeats to oneself inwardly ‘I-I’ with the entire mind fixed thereon, that also leads one to the same source.’

Tom:

It means to turn away from objects towards the Self; the mantra ‘I-I’ just being a means to point your attention Selfwards.

‘I-I’ means ‘I am I’ or ‘I am that I am’. This is because in most Indian languages ‘I’ and ‘I am’ are essentially the same word.

In the Path of Sri Ramana Part One it states on p.114:

The consciousness ‘I am’ when felt along with an adjunct (upadhi) as ‘I am so-and-so’ becomes a thought. Of all thoughts, this thought is the first. But the consciousness which shines alone as ‘I-I’ without any adjunct is Self (atman) or the Absolute (brahman). This is not a thought. It is our ‘being’ (that is, our true existence)

and on page 133 it states:

But as we have already said, it is to be understood that the consciousness mixed with adjuncts as ‘I am this’ or ‘I am that’ is the ego (ahankara) or the individual soul (jiva), whereas the unalloyed consciousness devoid of adjuncts and shining alone as ‘I-I’ (or ‘I am that I am’) is Self (atman), the Absolute (brahman) or God (iswara).

We can see that I-I (and the variants of this such as ‘I am that I am’, etc) simply refer to the Self, that is consciousness without any objects, as per the quote on page 133.