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I know this seems like a silly question but it was just a random thought that I had today.

I know Paladins only get so many spells a day by preparing them usually when they wake up and pray to their god. That much I understand but my thought was that there isn't anything saying they can't prepare the spells and then not use them a whole day and prepare more the next day - or is there?

My thought would be that a paladin can have a stockpile of spells being what ever he prepared the day before and spells he prepared the day after but can still only use the number spell slots he has.

For example I have a level 4 Paladin who can have 3 1st level spells prepared, but let's say I didn't use any of these spells for a whole day and prepare 3 more the next day. Would I have 6 different spells to use or would these spells just disappear?

The thought is he wouldn't be casting more spells a day (his limit being 3), he would just have more variety to use in certain circumstances. Does this seem feasible or is there something that says for sure that a paladin can't stock pile his spells?

The rules say change, not add to, prepared spells

The difference between "adding to those already prepared" (your proposal) and "changing those previously prepared" hinges on "the new list" being prepared after a long rest.

You prepare the list of paladin spells that are available for you to cast, choosing from the paladin spell list. When you do so, choose a number of paladin spells equal to your Charisma modifier + half your paladin level, rounded down (minimum of one spell). The spells must be of a level for which you have spell slots.

snip Casting the spell doesn’t remove it from your list of prepared spells.

You can change your list of prepared spells when you finish a long rest. Preparing a new list of paladin spells requires time spent in prayer and meditation: at least 1 minute per spell level for each spell on your list. (SRD. Page 31)

The new list replaces the old list; if the new list added to the previous list the rules would specify that. Note that adding to the previous list would run afoul of the numerical limit already presented (based on 1/2 level and Charisma modifier) om the Paladin spell casting section. (And for that matter, if it worked in an additive way, then in short order all spells on the entire Paladin spell list could be prepared, per @Upper_Case's comment, rendering any such limit rather pointless).

About 'using up spells' (title of question)

It doesn't matter if the paladin cast any spells the previous day or not.

What is "used up" in this edition when one casts a spell is spell slots. If I prepare three different spells - command, cure wounds, divine favor - and have three slots to cast for that day (per your example) I can cast command three times, or I could cast cure wounds once and divine favor twice, etc, in whatever combination comes up during play. The spells themselves are "fixed in the paladin's mind" (see Chapter 10) through preparation. I'd recommend a review of Chapter 10 (PHB or Basic Rules) in the sub sections of "Known and Prepared Spells" and "Spell Slots." (Italics mine in the excerpts below)

Known and Prepared Spells

Before a spellcaster can use a spell, he or she must have the spell firmly fixed in mind, or must have access to the spell in a magic item. Members of a few classes have a limited list of spells they know that are always fixed in mind. snip Other spellcasters, such as clerics and wizards, undergo a process of preparing spells. This process varies for different classes, as detailed in their descriptions. (Basic Rules, p. 82)

Paladins prepare spells the way clerics and druids do.

Spell Slots

Regardless of how many spells a caster knows or prepares, he or she can cast only a limited number of spells before resting. (Basic Rules, p. 82)

This is what gets used up between long rests by the paladin.