While many features are designed to scale with more characters, and thus are strongly recommended by alts, there really is only one thing that is considered a need for an alt: Capital ships.
Capital ships do not have a normal navigation system, while they can now use Stargates, doing that by yourself is very risky. Additionally many alliances in Nullsec do have Cyno beacons you can jump to freely, however they can still be trapped, or otherwise require a second character to scout.
To realistically be a capital pilot in any measure you require two accounts, one to be your cyno, and one to be the capital pilot. The nice thing is the cyno alt doesn't have to be active in anyway beyond lighting your cynos for personal movements, as many combat maneuvers will be done in a group, where a cyno will be provided. All that said having any capital ship without a cyno alt is basically asking to die eventually.
Other tasks that are generally considered to require an alt:
- Hauling values > 1 billion generally should have a scout, and unless
you have a real life buddy that you trust explicitly, this should
also be you.
- Any other tasks that require a scout such as lowsec or null hauling
- Any tasks in which scaling is an issue, this includes high end PI and Industry, and could arguably include mining
On the other hand, with the combat boosting changes coming in November, the playing field between alt assisted players and truly solo players is going to be significantly improved. So there is that.
As the other answer has said, technically most of this can be done by friends and other players, but EVE is about trust. And you never know what is on the other end of the jump unless you have eyes on it yourself.
Edit: Update - The changes to boosting did encourage solo gameplay. Also, the addition of solo content such as the Abyss has vastly increased solo play activities.