You can't refuse to level up...
From the Players Handbook, "Experience and levels" p 58:
Advancing a Level: When your character’s XP total reaches at
least the minimum XP needed for a new character level (see Table
3–2), he or she “goes up a level.”
(emphasis mine) That's not a something that you can choose to do or not. As soon as you reach the requirements, you're considered to be of the new level you reached. It's the same thing as aging. You can't decide to remain at 12 six years, then to directly go at 18.
Mechanically, that would probably break the game. The players would face encounters increasingly difficult, gaining increasingly more XP. He would potentially gain even more levels than what he expects. That seems pretty unfair to the others players, as they would have to carry him during the time he saves his points.
Furthermore, does he expect to show in front of the BBEG (lv 17, to accommodate the 15-th party level) and then snap his fingers to magically upgrade to level 17 and fight him on equal ground? That doesn't seems fair either, for the players and you.
... unless some very specific circumstances.
The very first entry about creation of magic items and XP cost is located in the Players Handbook, page 88:
XP Cost: Power and energy that the spellcaster would normally
have is expended when making a magic item. The XP cost equals
1/25 the cost of the item in gold pieces (see the Dungeon Master’s
Guide for item costs). A character cannot spend so much XP on an
item that he or she loses a level. However, upon gaining enough XP
to attain a new level, he or she can immediately expend XP on
creating an item rather than keeping the XP to advance a level
So, you can delay your progression, but not choose to stop it, and under specifics circumstances (e.g: being able to craft magic items, and deciding, when you hit the requirements, to craft an item instead). It does not allow you to voluntarily lose a level by spending too much XP, nor does it not allow you to save it. You gain it, you spend it on either, but you don't have a magical pouch of XP somewhere.
An "option" to lose a level, is for a wizard to get his familiar killed. The traumatism cause a XP loss, that can be halved on a saving throw (so not the loss is not done voluntarily). You can not "traumatize" yourself by crafting magicals items, as RAW.
As pointed by Fectin, the section "Experience and levels - Advancing a level" mention a case where you can't access a level:
A character can advance only one level at a time. If, (...) a character’s XP reward from a single adventure would be enough to advance two or more levels at once, he or she instead advances one level and gains just enough XP to be 1 XP short of the next level. Any excess experience points are not retained.
From what I understand, even in this case, you don't save XP, you merely get a level and the experience needed to get to the following minus one. So you level up. And the additionnal experience is lost. So you're twice as disappointed. No delay, no savings.