This is a tricky question. It depends heavily on what exactly counts as an activity or event. The obvious short answer is No. It's called the observable universe because it contains all that is observable by us. Anything outside of it is unobservable and, thus, can't influence us (else, we could use that influence to observe it). But how often is the obvious short answer the complete and correct answer? Correct; sometimes. But this is not one of those times. On to the long answer:
The reason I said it was tricky is because there are elements of the universe that are larger than our observable chunk that can influence us ("larger" may not be the right word. At least, they extend to scales larger than the observable universe). I'm going to present a few examples of things that may be considered as something outside the observable universe influencing us. Some of these are purely theoretical, some are from observations. This is not a comprehensive list and whether or not they count as examples is up to the reader (which is why I started by saying this is a tricky question).
The first example I could provide is a collision with a bubble universe. This is purely theoretical, but in the realm of eternal inflation, it's possible to have multiple "universes" (really, just regions that behave similar to our own universe) form out of the background and, if they form close enough together, you can get a collision between these two bubble universes (See below for a technical side note on bubble universes). This influences us in ways that we haven't fully explored, but we would expect to see it for the most part in the Cosmic Microwave Background (CMB). This would arguably be something outside our observable universe affecting us (or at least causing a theoretically observable effect).
Something a bit more probable is the primordial background perturbations in the universe. Before inflation (if it actually happened), there were inhomogeneities that influenced the flow of energy and the density of the universe. Inflation theoretically then expanded the universe to the point that these inhomogeneities were so much larger than the observable universe that they can no longer influence us. However, that initial influence remains in the density of matter/energy of our observable universe. We have no way to compare our region's flow and density to other regions, but if some of our best theoretical models are correct, we can be reasonably sure that these perturbations (not talking about the quantum fluctuations, which are smaller than the universe) have influenced us and most likely made us different from some other parts of the universe as a whole.
Okay, now let's talk about something we know influenced us. A long time ago, there were structures in our observable universe (galaxies, clusters, filaments, etc) that are no longer observable (expansion moved them outside the observable universe). These structures could and, in some cases, did have an affect on the flow of our galaxy cluster, the formation of objects around us, or even the matter content of our local region (not being an astronomer or a time traveler, I can't tell you exactly what one galaxy cluster 9 billion years ago did, please forgive me). Being outside the observable universe, these things no longer directly affect us, however, they can still indirectly affect us through the affects they had in the past. This is very similar to the "dark flow" discussed in one of the other answers. It's up to you whether or not that counts.
I know what you're thinking; "All those things are fairly sketchy and I'm pretty sure I want to argue with you about whether they are real and whether they actually count as things from the outside affecting us". Yes, I somewhat agree with that sentiment. This is why it's tricky. The issue is debated among cosmologists. You asked if anything outside the realm of what we can observe can affect us. Naturally, the answer isn't going to be "Yes, here's this example of something clearly affecting us" because then it wouldn't be outside the realm of what's observable. If you're willing to allow indirect effects, theoretical and subtle effects, or strictly non-measurable effects, then I can give you a yes. If you want to stick to significant and measurable effects with determined causes, then I'm going to have to give you a "No". Even dark flow is uncertain. We don't know for sure what the cause is, whether it is from outside the observable universe, a coincidence, or something else entirely. A tricky question. But I'm just here to provide information, it's up to you to choose if the information leads to a "Yes" or a "No" answer.
*Technical side note: We aren't talking about actual bubble shaped universes with a geometric center. The universes start as infinite in size, but the math is done is a frame where graphing it makes it look like a bubble. It is basically a hyperbola centered on a time-like axis but the space-like slices are themselves hyperbolas. At right angles to time, it looks like a bubble forming, but moments of equal time always show each universe as infinite.*