Rules Question: When Doubling Cube doubles mana, does that include the mana used to activate the ability?
No, Doubling Cube does not copy the mana used to activate it.
When you activate an ability you pay costs before the ability takes effect. Specifically, rule 602.2 describes the steps for activating an ability:
602.2a The player announces that he or she is activating the ability. [...]
602.2b The remainder of the process for activating an ability is identical to the process for casting a spell listed in rules 601.2b–i. Those rules apply to activating an ability just as they apply to casting a spell. An activated ability’s analog to a spell’s mana cost (as referenced in rule 601.2f) is its activation cost.
601.2h The player pays the total cost in any order. Partial payments are not allowed. Unpayable costs can’t be paid.
601.2i Once the steps described in 601.2a–h are completed, the [ability becomes activated]. Any abilities that trigger when a spell is cast or put onto the stack trigger at this time. If the spell’s controller had priority before casting it, he or she gets priority.
Normally, players would get a chance to respond before the ability resolves, but Doubling Cube's ability is a mana ability, so 605.3b says
An activated mana ability doesn’t go on the stack, so it can’t be targeted, countered, or otherwise responded to. Rather, it resolves immediately after it is activated. (See rule 405.6c.)
Then, rule 608 describes how spells and abilities resolve. Specifically, rule 608.2g says
If an effect requires information from the game (such as the number of creatures on the battlefield), the answer is determined only once, when the effect is applied. [...]
This means that the Doubling Cube checks the mana to be doubled when the effect resolves, which is after you spent the 3 mana on the ability's cost.