I have this makeshift file server made from an old laptop that doesn't support AC power back to automatically turn itself on when power is restored. I have to manually press the power button to start the server whenever a power loss occurs.
I do have an attiny13 at hand to do just this, but it honestly feels like a waste.
So, Is there any simple enough circuitry that basically connects the switch for about a second on power up?
the power button needs to be pressed for at least a second
Google 555 timer power up reset.
Use the CMOS version.
Drive a relay and use it’s dry contacts.
Yes, an ATTiny13 driving an optocoupler or reed relay.
Im trying to avoid using any chips. Also, the switch I'm pressing is the laptop power button itself, which carries a minuscule amount of current. So a relay would be redundant
You are totally missing the point, it gives you galvanic isolation.
However, sounds like you already have made up you mind. 
First, if you have arbitrary requirements like this, you need to state them upfront. Second, why? What's so special about chips? It's all just tools in use to solve a problem.
I mean, you could do it with an LC circuit, using the coil of a relay as part of a timing element, but you'll spend enough time tinkering with various capacitor values that I'd just throw a microcontroller at the problem and be done in ten minutes.
If you really want to pursue it, a large value capacitor like 3,300 uF in series with a relay will probably give you about a 1-second contact closure when power is applied. Determining the actual relay and capacitor values are left as an exercise for the student.
Why not just use a sufficient UPS and not have to mess with any circuitry??
I think you are also thinking to complicated. Just place a battery into the laptop. 
Not always so easy to find reliable batteries for old laptops, but a UPS is guaranteed to solve the problem and they are relatively inexpensive.
You're right on that one.
Its evident that im not an expert on electronics, and so I thought there may be other ways of doing this task. The timing doesn't need to be accurate, what I meant was that there was a minimum threshold that needed to be filled. Is it feasible to replace the relay with a transistor?
I saw a similar thing on another mini PC forum, and one proposed solution was by using a capacitor and resistor.the forum thread.
i’d go with a servo
That's exactly what I said would happen!
A 555 (or better, a microcontroller) and a relay would have had this problem solved already.
Just put a UPS on it and it won't glitch on power loss.
OP feels a $0.50 microcontroller and a $0.25 optoisolator is a waste. I doubt a UPS would cost less.
But we don’t know how it shuts down, it probably dies and needs disk check after every power failure
What does the server do?
Further, what is the cost of downtime?
yes, i finally settled with an attiny13. I think you misunderstand my definition of waste. I was looking for a separate solution, not the ones I already knew
its a file server, cost of downtime is none as i am the only user
