I got there with Paris Hilton after only 6 or 7 clicks.
If you allow for links in parentheses, esp. word origin, you’ll eventually get stuck in Ancient Greece. 
Philosophy is the most general discipline of enquiry, and embraces all others. As you may have heard, the word “philosophy” comes from the Greek meaning the love of wisdom (or, perhaps more accurately if less poetically, the love of knowledge), and, as originally practiced in ancient Greece it encompassed the search for knowledge about pretty much anything. Since that time, the search for knowledge about certain aspects of reality has become more successful and focused, such that many sciences, social sciences and humanities disciplines have branched off from philosophy and have grown greatly as enterprises. Nevertheless, they remain in essence, branches of philosophy, the original quest for knowledge in general that began in Greece. Thus any general encyclopedia is, fundamentally, a philosophical work.
If you start with any specific topic of enquiry (i.e. some random Wikipedia entry) and follow links randomly (which is essentially what you are being enjoined to do here - i predict that using every fifth link, or every seventeenth, would work just as well, just so long as you do not get to choose which link to click) each step will either take you to some more (or equally) specific topic, or to some more general topic. The more (or equally) specific topics will all be different (or show no particular pattern), but if you go more general for enough steps you will eventually arrive at the most general subject area of all, which is philosophy. Thus a random walk of this sort, if it does not get trapped in some loop, will inevitably eventually lead you to philosophy.
There’s actually a mathematical reason why most entries leads to a single loop of entries. Look at the following book:
First, define a random function. Suppose we took all the integers from 1 to n. At random choose one of the other integers in that range to be mapped to. Every integer has to be mapped to a single integer, but not every integer is mapped to. Suppose, for instance, the function is the following on the integers from 1 to 13:
1 -> 4
2 -> 11
3 -> 1
4 -> 6
5 -> 3
6 -> 9
7 -> 3
8 -> 11
9 -> 1
10 -> 2
11 -> 10
12 -> 4
13 -> 7
You can draw a graph of this. Look at the top of page 55 in the book I’ve linked to. There is the graph of the function I have given above.
As is defined in that book, we can talk about the number of components in the graph. Each component is a cycle and all the points that lead to the cycle. So in that graph at the top of page 55 there are two components. The component on the left has nine points (i.e., integers) in it. The component on the right has four points in it.
Now suppose you took a large number of points and hooked them together in a random graph. There are currently (as of this second, so don’t complain to me that when you read this post it’s greater) 3,649,553 entries in Wikipedia in English. Every entry is mapped to one other entry according to what the first link in the entry is. So you can create a graph of this function. The thing is that it doesn’t matter whether you create the function by mapping each entry to the first link in the entry or just to some random other entry. According to Fact 2.37 in the book, you can expect the largest component in the graph you’ve created to be about 2/3 the number of all the points in the graph.
For English-language Wikipedia entries, we thus expect the size of the largest component to be 2/3 * 3,649,553 = 2,443,035 1/3 points. So there’s nothing surprising about the fact that most entries lead to a single cycle.
[O.K., I know somebody is going to complain about the following: This can’t actually work like this was a random function and a random graph. In a graph formed by a random function, the cycle in the largest component should be much larger. Yeah, O.K., it’s doesn’t quite work to treat this as just a random function.]
I’m not quite convinced that this is the right explanation. The graph that describes all of Wikipedia’s internal links is a scale-free network, which seems to be very different from the sort of graphs you’re addressing. The subgraph generated by the first links is a spanning tree, and probably nearly minimal if you give all the edges equal weight. Minimal spanning trees of a scale-free network are intuitively very likely to pass through nodes with very high degrees, and I expect that that’s what we’re seeing here. I’ll run it by some friends of mine who work in network science and see what they have to say.
You, and every other male in the western hemisphere.
OK, it’s definitely not a tree, but it’s a random subgraph that looks a lot like a tree. The randomness is why not every article works.
I keep getting to Latin and Greek. The latter is a closed loop, and the former itself leads to the latter. Am I not supposed to be skipping the disambiguation and see also links?
Yes, ignore parenthetical and italicised links.
Ah. It works now. From “penguin” at least. I was ignoring italicized stuff, but I didn’t notice the bit about parentheticals.
ETA: “Kevin Bacon” leads to a loop involving “recording” and “film”.
Out of curiosity, I tried following the second non-italicized, non-parenthetical link in every page. The first try got me into a two-element loop of “word” and “semantics”. My second try was the same loop, but the third try got into a longer loop of organism -> contiguity -(skipping over dead link)-> time -> sequence -> set (mathematics) -> mathematics education -> learning -> behavior -> organism. There’s also a loop with classification <-> taxonomy, one with National League <-> National Association of Professional Base Ball Players, and probably several others. The Word <-> Semantics loop is probably the most common destination, though.
Incidentally, while checking random links, I also found an exception to the philosophy rule: The first ten links on the 1750s page are all to anchors on the same page, making it a self-loop.
From Front Page -> Casino Royal (2006 film) it took 32 links. I kept getting close to Philosophy, before veering away from it. That’s not including terms that had philosophy in parentheses, including object, property, state and event.
It’s probably best to ignore anchor links. The first page link (1750) does rapidly lead to philosophy, but you do have to skip over some Wiktionary links.
I’m going to experiment with this variant. Already, with my first attempt (Apollo 13) clicking every 5th link leads to the loop of Biloxi, MS -> Harrison County -> Biloxi.
It seems to me that what all these links actually lead to is “Property (Philosophy)”. The first link on that page is to modern philosophy so you automatically get there once you’ve hit that page, but it’s “property” that you really get led to.