Are there any drinks which upon consumption a person would make the blessing of boreh pri HaAdamah?
There is an opinion that if certain drinks are processed by heating/cooking they would warrant the brocha of ha'adama as through this process it extracts more of the original flavour.
The Chayei Adam 52:1 explores this concept. He writes there:
דין רוטב של פירות וירקות (סי' ר"ב)
המבשל וירקות לאכילה מברך גם על הרוטב בפה"א דאע"ג דעל מי פירות כשסחטן מברך שהכל כדלעיל כלל מ"ט סי' ג' י"ל דיותר יוצא טעם הפירי ע"י בישול משיוצא ע"י הדריכה והסחיטה
One who cooks vegetables to eat, makes the 'ha'adama' blessing also on the sauce/juice since even though with the juice of the fruit when it is squeezed, one makes a 'she'hakol' as we stated earlier klal 49, se'if 3, there are those who say that there is more of a taste of the fruit when extracted through cooking, more so than squeezing...
So at least according to this view, something like tomato juice, if it has been heated/cooked as part of its production would require the brocha of ha'adama.
This notion seems to be supported by Rav Tzvi Pesach Frank in Har Tzvi here (left column, second paragraph) who has a question on the production of cognac. As an aside, he looks at the Gemara in Brachos 38a that talks about date honey and how it is a brocha of 'she'hakol' as it retains no essence of the actual fruit but merely the moisture. However in the proceeding Gemara in Brachos 39a it talks about the water in which beets had been boiled and is regarded as having the same status as the beets themselves. He bring the Tosfos there where it writes:
"We recite on the beet soup בורא פרי האדמה even though the soup has in it only the juice of the beet and its taste; one recites a b’rochoh upon the soup as he recites on the vegetable itself. Even though we have learned earlier (38a) that fruit juice is merely sweat of the fruit and not fruit itself and we therefore recite שהכל נהיה בדברו, one can differentiate between the two."
So once again, if a vegetable-based drink is heated as part of its production it retains the same brocha as the vegetable.
As an update about this question:
Recently Rav Dov Lior published a shiur (available here: https://www.youtube.com/watch?v=veFtVA9dnh4) in which he says that fruits/vegetables which producers produce with intent of the production going at least partially for the making of juices (אדעתא דהכי), juice made from them, if there is no other things mixed on it, remain with it's original blessing. For example: carrot juice would require "HaAdamah".
He says that's Rashba opinion and the Chazon Ish pasken like it (12:09 in the video).
Disclaimer: This does not answer the question for HaAdamah blessing but only presents a related case in which this happens for haEtz blessing.
Rav Ben Tzion Aba Shaul writes in his shut Ohr Letzion chelek 2 Perek 14 Piska 7 that if someone would squeeze the juice out of a grape directly into his mouth the beracha would be haEtz (just becomes haGafen when it gets to a container/cup). He writes on the notes that this is only true for grapes/wine, all other fruits even when squeezed directly into someone's mouth become shehakol.