I've sorted the options which I've found and deemed reasonably close by increasing level, pushing spells towards the end.
The Scourge Aasimar's Radiant Consumption trait (Volo's Guide to Monsters, p. 105) lets them:
[...] unleash the divine energy within yourself, causing a searing light to radiate from you, pour out of your eyes and mouth, and threaten to char you. [...] During it, you shed bright light in a 10-foot radius and dim light for an additional 10 feet, [...]
The other Aasimar subraces have similar activated abilities, though they are glow-y eyes and wings, rather than an actual aura. (Thanks to By Symmetry for pointing this one out.)
Path of the Storm Herald
When raging, a barbarian who follows the Path of the Storm Herald (XGtE, p. 10):
emanate[s] a stormy, magical aura while [they] rage.
That aura is desert-, sea-, or tundra-themed depending on the environment they choose, granting additional benefits as they increase in level.
Way of the Sun Soul
Also from XGtE (p. 35), Monks who follow the Way of the Sun Soul until 17th level gain the Sun Shield feature:
You shed bright light in a 30-foot radius and dim light for an additional 30 feet.
They can activate or deactivate the light as a bonus action. The "aura of power" aspect fits even better because they can deal radiant damage as a reaction if someone hits them while they're glowing with this aura.
Oath of Devotion
A Paladin who has sworn the Oath of Devotion and reached 20th level gains the Holy Nimbus feature, which says that:
as an action, you can emanate an aura of sunlight. For 1 minute, bright light shines from you in a 30-foot radius, and dim light shines 30 feet beyond that.
There's also some radiant damage for enemies that start their turn in the aura, and advantage for you on saving throws against spells cast by fiends/undead.
The 8th-level cleric spell holy aura opens with:
Divine light washes out from you and coalesces in a soft radiance in a 30-foot radius around you.
I'll just here mention that the other aura spells (aura of life, aura of purity, aura of vitality - all on PHB p. 216) have you emanate life-giving energy or something similar, which are not explicitly described as visually apparent. Other spells like crusader's mantle (PHB, p. 230) have similar wording.
There are, on the other hand, a number of spells which add some visual up-to aura-like effect on you (or in some cases other casters). I've generally ignored spells where the effect is described as faint or similar; we're kinda looking for something obvious here. I also can't guarantee that this is a complete list.