She's hard to type, in the interviews I have watched she presents a different side to her everytime. In recent interviews she is quite different to earlier interviews when he career had just started to take off.
I'm a huge fan of hers and I have tried to type her many times but I haven't settled on one type. It's hard to type somebody I've only observed and not interacted with. From the interviews I have watched (I'm not taking her songs and her performances into account, just the person) this is what I have:
She appears to be very abstract, very much in her own little world. She's clearly very imaginative, a visionary. If you look at her eyes in some of the interviews (I can't find the link to the ones I am talking about. When I do I'll edit this post) she looks as if she is disconnected,...I imagine she often feels like she's on the outside looking in most of the time. She appears to be more of an idealist, I don't think she has a strong grip on reality. I think she is an intuitive. I think she has Ne, but I don't believe it's her dominant function. I think it is her auxiliary function as that is what she presents most to the world but Fi is her dominat function. She only seems to present her Fi outwardly in situation where she is comfortable and it matters to her "inner world" e.g like in the interviews with Ellen and Oprah. In some interviews she doesn't present this, sometimes giving very short answers.
This would confirm what I think anyways...that she is an introvert. So many people think she is an extravert because of her performances. They are confusing shyness and introversion. She is clearly introverted. You have to remember, she went to a convent school and wasn't really allowed to express herself. She felt like a freak, I haven't watched or read anything where she goes into more detail about this but I can easily imagine she was bullied for being so artistic and unconventional. Because of the environment she wasi n she must have really wanted to rebel. And that's what she did because she really feels the need to express herself...and that's what comes out in her performances. I don't believe she is an attention-seeker. I also don't believe she is stimulated by social interaction, she seems quite awkward around people. She has said she is a perfectionist and works really hard not only with her music but also with the set designs, outfits etc and spends a lot of time working on them. She has said she doesn’t go to nightclubs to socialise (or words to that effect).She has said that the biggest misconception about her (in the Barbara Walters interview) is that she is an attention seeker. She is extremely creative and needs to express herself. She may do it in ways that are "different" and sometimes provocative but that doesn't make her an extravert. I'm an introvert but I am also shy. If I wasn't shy I can't definately say I wouldn't express myself in the way she does. I wouldn't describe her as shy, but that doesn't make her an extravert. And not forgetting the more obvious signs, like how softly spoken and reserved she is in a lot of interviews.
She has said many time her fans are important to her and she clearly has such a depth of caring for them, she even cried in an interview (I think it may have been the Oprah one). This leads me to think she is not a thinker. I'm NOT saying thinkers don't cry before anybody takes that the wrong way. She appears to make decisions by following her heart and I think she is more subjective than objective.
(From wikipedia) -Fi filters information based on interpretations of worth, forming judgments according to criteria that are often intangible. Fi constantly balances an internal set of values such as harmony and authenticity. Attuned to subtle distinctions, Fi innately senses what is true and what is false in a situation. I think she definately has Fi. If she had Fe she would care more about acting appropriately and fitting in with social norms.
We also have to take into account what all these rumours about her gender are doing to her personality. She might not appear as if she cares, but it must be so hurtful for people to say you look masculine....to doubt your gender altogether. She felt like a freak at school, so it all must be bringing back a lot of painfuly memories. And a lot of interviewsers bring this up so it must make her feel uncomfortable...which will have an impact on the way she comes across.
From all this I get INFP. Based on the interviews I have watched I think that is a good fit. I'm not saying this is correct, I don't know her so it's almost impossible to say. I have doubts about INFP so feel free to disagree. I'm not an MBTI expert and like I said earlier I'm making this guess at her type from obervations, not social interactions. The only thing I am certain of is that she is not an NT like some people have said. I can see why people have come to that conclusion but I just don't feel it myself. And when people said she was a J...I feel she is too spontaneous to be a J and too unconventional. I don't believe she is a planner, I think she decides what to do as she goes a long...as new ideas enter her head. She is extremely creative so she probably likes to leave a lot of things open for new possibilities. Plus taking into account what I said earlier, about Ne. That is her percieving function and is is extraverted so she must be a P. I feel she presents Ne to the world more than Fi.
But feel free to tell me this is a load of rubbish

I'm very intersted in Lady Gaga and would very much like to know her type so anything you wanna add would be great
EDIT: Just thought about it. Having major doubts about what I just wrote. I'm trying to think through it logically...but logic isn't my strong point.
I'm leaning towards ISFP. I'm not saying she's an intuitive because she's a little strange...sensors can be strange too (believe me, my father is ESFP...I should know

).
When she first came onto the scene I thought she was most likely an ISFP. Maybe I should stick to my first instinct? This means she would have Se.
Se focuses on the experiences and sensations of the immediate, physical world. With an acute awareness of the present surroundings, it brings relevant facts and details to the forefront and may lead to spontaneous action. I'm actually starting to think it's possible I was wrong.

Plus I don't know enough about the sensing/intiuting preferences to be able to judge.

Oh and I also think she is type 4. I agree with other posts that say she is 4w3.