Bilbo DID become addicted to the Ring - when it came time to part with it, he had a full on freakout:
'Everything?' said Gandalf. 'The ring as well? You agreed to that, you remember.'
'Well, er, yes, I suppose so,' stammered Bilbo.
'Where is it?'
'In an envelope, if you must know,' said Bilbo impatiently. 'There on the mantelpiece. Well, no! Here it is in my pocket!' He hesitated. 'Isn't that odd now?' he said softly to himself. 'Yet after all, why not? Why shouldn't it stay there?'
Gandalf looked again very hard at Bilbo, and there was a gleam in his eyes. 'I think, Bilbo,' he said quietly, 'I should leave it behind. Don't you want to?'
'Well yes – and no. Now it comes to it, I don't like parting with it at all, I may say. And I don't really see why I should. Why do you want me to?' he asked, and a curious change came over his voice. It was sharp with suspicion and annoyance. 'You are always badgering me about my ring; but you have never bothered me about the other things that I got on my journey.'
'No, but I had to badger you,' said Gandalf. 'I wanted the truth. It was important. Magic rings are – well, magical; and they are rare and curious. I was professionally interested in your ring, you may say; and I still am. I should like to know where it is, if you go wandering again. Also I think you have had it quite long enough. You won't need it any more. Bilbo, unless I am quite mistaken.'
Bilbo flushed, and there was an angry light in his eyes. His kindly face grew hard. 'Why not?' he cried. 'And what business is it of yours, anyway, to know what I do with my own things? It is my own. I found it. It came to me.'
'Yes, yes,' said Gandalf. 'But there is no need to get angry.'
'If I am it is your fault,' said Bilbo. 'It is mine, I tell you. My own. My precious. Yes, my precious.'
And there we go. "My precious".
The wizard's face remained grave and attentive, and only a flicker in his deep eyes showed that he was startled and indeed alarmed. 'It has been called that before,' he said, 'but not by you.'
'But I say it now. And why not? Even if Gollum said the same once. It's not his now, but mine. And I shall keep it, I say.'
Gandalf stood up. He spoke sternly. 'You will be a fool if you do. Bilbo,' he said. 'You make that clearer with every word you say. It has got far too much hold on you. Let it go! And then you can go yourself, and be free.'
'I'll do as I choose and go as I please,' said Bilbo obstinately.
'Now, now, my dear hobbit!' said Gandalf. 'All your long life we have been friends, and you owe me something. Come! Do as you promised: give it up!'
'Well, if you want my ring yourself, say so!' cried Bilbo. 'But you won't get it. I won't give my precious away, I tell you.' His hand strayed to the hilt of his small sword.
Gandalf's eyes flashed. 'It will be my turn to get angry soon,' he said. 'If you say that again, I shall. Then you will see Gandalf the Grey uncloaked.' He took a step towards the hobbit, and he seemed to grow tall and menacing; his shadow filled the little room.
Bilbo backed away to the wall, breathing hard, his hand clutching at his pocket. They stood for a while facing one another, and the air of the room tingled. Gandalf's eyes remained bent on the hobbit. Slowly his hands relaxed, and he began to tremble.
'I don't know what has come over you, Gandalf,' he said. 'You have never been like this before. What is it all about? It is mine isn't it? I found it, and Gollum would have killed me, if I hadn't kept it. I'm not a thief, whatever he said.'
'I have never called you one,' Gandalf answered. 'And I am not one either. I am not trying to rob you, but to help you. I wish you would trust me, as you used.' He turned away, and the shadow passed. He seemed to dwindle again to an old grey man, bent and troubled.
Bilbo drew his hand over his eyes. 'I am sorry,' he said. 'But I felt so queer. And yet it would be a relief in a way not to be bothered with it any more. It has been so growing on my mind lately. Sometimes I have felt it was like an eye looking at me. And I am always wanting to put it on and disappear, don't you know; or wondering if it is safe, and pulling it out to make sure. I tried locking it up, but I found I couldn't rest without it in my pocket. I don't know why. And I don't seem able to make up my mind.'
'Then trust mine,' said Gandalf. 'It is quite made up. Go away and leave it behind. Stop possessing it. Give it to Frodo, and I will look after him.'
... and yet, he STILL doesn't part with it:
'You have still got the ring in your pocket,' said the wizard.
'Well, so I have!' cried Bilbo. 'And my will and all the other documents too. You had better take it and deliver it for me. That will be safest.'
...
And later on, Gandalf confirms it to Frodo:
'A Ring of Power looks after itself, Frodo. It may slip off treacherously, but its keeper never abandons it. At most he plays with the idea of handing it on to someone else's care – and that only at an early stage, when it first begins to grip. But as far as I know Bilbo alone in history has ever gone beyond playing, and really done it. He needed all my help, too. And even so he would never have just forsaken it, or cast it aside. It was not Gollum, Frodo, but the Ring itself that decided things. The Ring left him.'