How many times are we required to pray daily, besides our morning prayers ("after waking up") and before going to bed?
There is no biblical instruction about the frequency of prayer. The exhortations in the New Testament to "pray without ceasing" (1 Thess 5:17), probably means simply not to desist from prayer (and see Paul's statement in Rom 1:9 that "without ceasing" he remembers the Roman Christians in prayer.
The portayal of Daniel praying three times daily in Daniel 6:10 may reflect a practice followed by some ancient devout Jews, but there is no commandment to do so either in the Old Testament or New Testament.
Prayer is probably approached as something that one does for positive purposes of communion with God, to express praise/thanksgiving, to offer petitions and intercessions, etc., and not a regulation to be complied with a certain number of times daily.
Conceived as personal conversation and spiritual communion with God, prayer is above all the joy and the light of the soul. However, for many of us, until we grow in the life of prayer, it is also a challenge and a task requiring discipline and set times of Bible reading, reflection, self-examination, prayer, asking for forgiveness and guidance, which can be accomplished in early morning or evening or both, as deemed helpful. Part of the goal of such scheduled times of prayer is to acquire the spirit of prayerfulness to retain during the day, together with brief lifting up of the mind and heart to God at various moments during the day. A growing believer ought to thank God at each meal. At special times of need, we have the example of Jesus who prayed three times on Gethsemane night and of St. Paul who prayed three times to be healed of an ailment (2 Cor 12:8). We must, however, be prepared that the answer might be "no" and go faithfully on to our tasks as both Jesus and Paul did. Not enough could be said about the meaning and value of prayer which the saints equated with true life with God. Let a quote from St. Theophan the Recluse, a Russian saint of the 19th century, suffice: "When prayer is right, everything is right."
We know from Scripture of many different times of the day in which people prayed to great effect. The concern, however, is not to mandate exact times or a certain number of times, but with the motivation, content, and results of prayer. 1 Thessalonians 5.17 says to "pray without ceasing." This is not intended to be taken literally. We then could not carry out other tasks that God has given us to do. It does set forth the ideal that we should strive to actualize being in constant awareness of God and God's love and care of us and of our love and loyalty to God so that we readily and spontaneously speak to God and sense God's speaking to us. Morning and evening prayer, even if brief, provide a helpful framework for that constant spirit of prayer.
As "Christ is the end of the law" (Rom. 10:4), so we are not "required" to pray any sent number of times per day at all. But as "true worshipers...worship the Father in spirit and in truth" (John 4:23), so even one truly heartfelt prayer a day is better than 100 rote, thoughtless repetitions per day.
While many of us have been raised to pray in the morning and before bed, the Bible does not actually contain any such requirement. Paul does encourage the Thessalonians to “rejoice always, pray without ceasing, give thanks in all circumstances” (5:17); however, this is less about a prescribed practice than the cultivation of an attitude: i.e., we should constantly direct our thoughts towards God, seeking God’s guidance in all that we do. Paul himself exemplifies this. It is noteworthy that Paul begins nearly all of his letters with a prayer of thanksgiving for those whom he is addressing. Jesus, when asked by the disciples, teaches them what has come to be called “the Lord’s Prayer” (Matt 6:9-13; Luke 11:2-4). Yet this too is more of an outline – suggestive of the way in which we should pray. He does not specify when or how often we should pray.
Paul says "Rejoice always, pray constantly, give thanks in all circumstances; for this is the will of God in Christ Jesus for you." (I Thess 5:16-18). Christian tradition-- especially the monastic strand-- takes this counsel seriously. The liturgy of the hours, for example, (also known as the "divine office") includes prayer seven times during the day and night, underscoring the idea that all of one's life ought to be dedicated to a constant dialogue with God.
Practically speaking, most people are not monks, even though I believe we can learn a lot from monastic practice. It is good to engage in formal prayer every day, even if it's the short thanksgiving before a meal, or praise at the beginning or end of the day. Liturgical prayer, too, joins us to the wider family of God, and this too is a good regular practice-- for example, as a way of keeping the sabbath holy (cf. Ex 20:8 and Dt 5:15).
But there are many more informal ways to pray: uttering the name of Jesus in the course of the day; offering an intention for a beloved; asking for God's help in the midst of a tiresome project; and so on. In this way, nearly any activity can be caught up in prayer "constantly," as Paul says. Prayer is conversation with God, said St. John Chrysostom; this idea of "conversation" has the same root as "conversion," i.e. "turning together." it's a wonderful image of what happens in prayer, of our turning our attention, our hearts and minds, with God toward the ordinary work of the day.
Traditional Judaism requires 3 times a day of rather longer, formal prayers, plus a few short prayers on waking and before bed.
But these are rabbinical rather than biblical commands. And apply to men only.
In the Torah the command is to sacrifice, not pray. Later in the Bible there are accounts of prayer (e.g., Hannah praying in the Temple to become fertile) but as far as I know no explicit rules until Talmudic Judaism replaces Temple-based sacrifice. The 3x a day prayers are, in fact, meant as a kind of replacement for sacrifices rendered impossible by the destruction of the Temple.