Scanned documents are generally just as good as paper copies. There are a few things that you might want to keep around indefinitely:
1) For investments in taxable accounts where your brokerage doesn't have to track the cost basis for you (generally shares that you bought prior to 2011, but there are exceptions), keep statements showing the purchase price until you sell the shares, and for a few years afterwards in case you get audited.
2) Keep records of money you put into your post-tax retirement accounts. If you ever decide to take early Roth distributions, you'll need this documentation to prove what fraction of your IRA balance comes from contributions vs. conversions vs. earnings. For IRA contributions your brokerage will send you a Form 5498 after tax season. For Roth 401(k) contributions those amounts will be shown in box 12 of your W-2 with code "AA". You'll be reporting any Roth conversions on Form 8606, so keep your tax returns from years when you do this.
3) If you have an HSA, keep your medical bills for any procedures you have after you open the HSA. When you make withdrawals you'll need to show that your medical expenses were at least as high as the amount you withdraw, and you could conceivably need to show that those expenses were not claimed for HSA withdrawals in a previous year.
There are probably other types of documentation worth keeping forever, but these are the ones that I personally plan to need.
