Of course there is a set order in the Siddur, but is there a hierarchy in ברכות השחר? If, for example, someone did not wash his hands or use the bathroom when he woke up, can he say Elokai Neshamah or ברכות התורה?
One pair of brachot in the morning that must be said in order is Mattir Asurim before Zokef Kefufim. If one said Zokef Kefufim first, he no longer recites Mattir Asurim (Shulchan Aruch OC 46:5).
Some say the set of Shelo Asani X's must be said in order, and if one said a later one he can no longer say the previous ones. However the Mishna Berura who quotes this opinion (46 sk 16) tends to reject that position.
Some require Elohai Neshama to be lechatchila recited after another bracha because it doesn't itself start with "Baruch Attah...", but others reject this requirement (see OC 6:3 with commentaries). All agree this would not invalidate the bracha.
That's all I can think of in terms of required orderings. Different Nuschaot have different traditions for the rest of the order.
To add to DoubleAA's answer:
The brachah of Ahavas Olam may exempt you from Birkas HaTorah. Therefore, you should say the latter first (47:7). The brachah of Mechayah HaMeisim may exempt you from Elokai Neshamah (MB 6:12). DoubleAA wrote in his answer that some say that you have to say Elokai Neshamah after Asher Yatzar; the Rama says to say Birchos HaTorah immediately after it. The korbanos have a specific order, as explained in 1:5-8 (and see Aruch HaShulchan there for the explanation of the Ashkenazi practice of saying only three things). I thought I remembered seeing in the Mishnah Brurah that you can say the brachos even if you don't experience the things (e.g. hearing an actual rooster crow), but I can't find it.
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