Definitely an odd film, it is best to take it as a parody of the spy-film genre: as such it is enjoyable. Michael Caine is mostly sort of half bemused and half confused as the hapless Harry Palmer whose job is drawing him deeper into insanity and mayhem. And implausibility. The culminating scene is, well, pure symphony of the best (read:trash) special effects of the day. The plot is full of twists and double-crosses, and includes a Texan bent on taking over the world (how very now).
If you are Finnish, or have visited Finland, the experience is either heightened or or lowered: Billion Dollar Brain is one of the films where Finland stands as a location-double for the unaccessible Soviet Union. It is hard to concentrate on the plot, when first Helsinki is playing Helsinki, then Porvoo is in Russia, and Riga is again in Helsinki. The border is seemingly in Hameenlinna. One ends up wondering how Harry does not realize his train is going merely back and forth. Location-spotting can keep you amused as well, though.