Unlike the first Diablo and Diablo II, players will be given a finite amount of Healing Potions in Diablo IV. That said, players will be able to increase the amount of Healing Potions they can carry at any time. This is done by increasing your Renown in any given region. For example, upon reaching a certain amount of Renown in the Fractured Peaks, players can claim rewards through the Renown tab to increase the amount of Healing Potions they can carry in Diablo IV.
Increasing Renown is done through an array of activities. This can be through activating Waypoints, which is Diablo's version of fast travel, discovering areas (or just exploring the map), finding Altars of Lilith, completing Side Dungeons, finishing Side Quests, and taking on Strongholds. Thankfully, before you throw yourself at Strongholds, you can easily up your Renown by simply walking around and exploring. But if you want to bolster the amount of Renown you get quickly, it is also advised that you pick up any Side Quests you see in any of the towns and cities you will come across.
Upon reaching Renown Rank II in Diablo IV, you will then be able to increase the amount of Healing Potions you can carry. In general, it's good to increase your Renown in a region for these optional rewards, as it will net you extra experience, gold and Obols. Obols are the currency used to obtain Whispering Keys or even Legendary items at random from the Purveyor of Curiosities. Paragon Points are also a reward you can obtain, which will be essential to end game builds.
And don't forget that Healing Potions can also be recharged by killing monsters. So don't be afraid to use them if you see a monster drop one. These Healing Potions appear as red orbs, and they will recharge exactly one use of your total amount of Healing Potions.