to give an example of... what? without theory there is no definition of the word intuition. i could give all sorts of examples of various moments of cognition that strike me as odd, but the more they are odd, the less likely they are related to type.
being intuitive about thoughts: meaning you know there is something wrong with your theory but you can not justify or explain how it is wrong. something in you seems to want to believe the opposite. you may have a clear idea about the other version, but cant justify it either. you live with the doubt and entertain both possibilities, even if that means to have doubt about the most significant aspects of your identity of reality. the cause of your doubts may not be related to the topics of your thoughts. rather your psyche is "abusing" the topic to communicate something else to you. everyday topics like typology become like dream symbols that have two meanings: their original objective meaning, and their meaning to the subject - that's you. but the two processes inform each other and must be either synchronized or differentiated in a meaning full way. because you are not just yourself, your brain is the intelligence that creates your whole world and all ideas about it. if you would try to be "objective" you would try to be reductionist, that is to say you would minimize the field that you deal with until you would foolishly believe, that it's too small for your subject to enter it and distort it. introverted perception has no interest in limiting your view in such a pragmatical way because it is not interested in practical reproducible manipulations. it wants to maximize the view and include more and more of the subject and thus embrace the unpredictable, the future of understanding. i could never give an example of this, because its a very long and complicated process of creating your understanding, your view of self and world. it's as complicated as i am. or as complicated as the world. obviously everyone is creating and maintaining a worldview that unites subject and object. this image or map is essentially what we are. but for some people it happens in the background, automatically. i push it around, semi-consciously, fully intentionally (curiously), collecting bits, tweaking tiny aspects of my shrine at a time.
being intuitive about specific sensory input: allowing objects to open you to imagination. this is the perception of art. you hunt through a collection of random objects (say wikipedia articles about vikings, funny words like "kimbern", some nice pictures) and observe what parts of yourself are highlighted in yourself. those parts are the puzzel that you try to get together. true vikings may not be that interesting to you. you are only looking for the viking in your self.
being intuitive about patterns and mapping them with meanings: this is mostly spacial intelligence and association, meaningful memory access. may not be related to typological intuition. but it provides instant insight into the nature of some entity, so it is likely motivated by the personality of a perceiver or intuitive. example: physiognomy. but patterns are not just in visual items, but all sort of abstract topics, but according to the way i understand the term "spacial intelligence" it's what connects things in ways that are not logical. however the connected things may be logical objects. so reading between lines or theory of mind (identifying cognitive functions) requires a spacial intelligence that maps associations and memories.
for me it also seems to be this spacial mapping that creates understanding of movement, process, intrinsic character. its the purpose of the mapping. as soon as i have the mapping i can predict the process. so the map is a function. its somewhat alive. before that i am only "intuitive" about how to complete a map that i am looking for.
the above is inspired by my introverted perception (Ni, Si, i don't care much to separate them anymore) but my wordings may be more general or imprecise ...
being intuitive about whats going on in general: asking for inner guidance, recognizing synchronicity. understanding how live works. too big of a topic to give a specific example. this original meaning of intuition is more like a stage of spiritual development and it's probably somewhat unrelated to typology. its the sort of intuition that people don't listen to. but to the degree you listen to it, it may have something to say about everything. its just to irregular to be typological function. example: my mother was lost. she has a bad case of alzheimers. there was absolutely no way to calculate her whereabouts. i knew where she was. it was a random place. i had no reason to know. i did consciously chose to not listen to it. i had a inner conflict about it. i decided to search town in a effective grid pattern instead. she was found by strangers at the random place i had in mind, initially.