While there are various proofs for God's existence, I've found this to be the simplest and therefore the most compelling. If you can refute this, let me know. Seriously, I believe this sums up the claim in Romans 1:19-20 and is undeniable. I'm sure others have expressed this argument, but here it is in my words:
For anything to exist, something must have always existed. Or to put it another way, if there was ever a point at which absolutely nothing existed, there could only be absolutely nothing now. So something exists eternally.
The physical world is decaying and will over time be completely cold and motionless, so the "eternal something" is not physical matter. If the physical world were eternal, it would have completely decayed "an eternity ago".
This "something" is self-existent, eternal, and transcends our physical/material order and is commonly and historically referred to as "God". Other names are "Prime Mover", "First Cause", "The One", etc.
To deny the existence of "God" as defined here is contrary to evidence and reason.
God as defined here is still a thousand miles from the God of the Bible, but this is the sure foundation that Romans 1 mentions. From here, the obvious action is for us creatures to seek out this eternal God and be thankful to Him for creating us. We should seek to know why He made us and if He has a further purpose for us. If we turn aside and worship created, non-eternal things it's obvious we are denying the Creator's special honor and should not be surprised to feel His wrath.
Another proof that is almost as irrefutable is that of moral necessity, as argued by Arthur Leff in his classic 1979 address at Duke Law School. Leff argues conclusively that there is no possibility of a definite moral standard without God. As a committed agnostic, Leff embraces the alternative to God – that there is no moral standard. He states "everything is up for grabs", including the idea that "napalming babies is bad" or that slavery is wrong. I contend that no rational person can truly embrace the idea of no morality or live consistently with that idea even if they say they mentally assent to it.
Romans 1:18-23
For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who suppress the truth in unrighteousness, because that which is known about God is evident within them; for God made it evident to them. For since the creation of the world His invisible attributes, His eternal power and divine nature, have been clearly seen, being understood through what has been made, so that they are without excuse. For even though they knew God, they did not honor Him as God, or give thanks; but they became futile in their speculations, and their foolish heart was darkened. Professing to be wise, they became fools, and exchanged the glory of the incorruptible God for an image in the form of corruptible man and of birds and four-footed animals and crawling creatures.
A search for God, grounded in thankfulness and "creaturely" humility, would not be properly satisfied until it finds the mercy of God through Jesus Christ. There are surely many distractions and selfish, sinful tendencies that can thwart a search for God. It is only by His mercy and the call of His Spirit that any of us are able to find and understand the mercy He offers us.
See Exit Strategy for more on why Jesus Christ is the only proper end to any true search for the "eternally powerful divine nature" who we are all inescapably aware of.