I think the usual Scriptural reference to "pray always" is 1 Thessalonians 5:17, which appears in the middle of this passage:
We urge you, brothers, admonish the idle, cheer the fainthearted, support the weak, be patient with all. See that no one returns evil for evil; rather, always seek what is good (both) for each other and for all.
Rejoice always. Pray without ceasing. In all circumstances give thanks, for this is the will of God for you in Christ Jesus.
Do not quench the Spirit. Do not despise prophetic utterances. Test everything; retain what is good. Refrain from every kind of evil.
Notice the overall tone of the passage, the last words of the letter before the final benediction. In broad strokes, St. Paul covers the sorts of day-to-day things Christians should do. "Pray without ceasing" is one of more than a dozen general instructions.
Ephesians 6:18 contains a similar instruction, but the context is different:
With all prayer and supplication, pray at every opportunity in the Spirit. To that end, be watchful with all perseverance and supplication for all the holy ones and also for me, that speech may be given me to open my mouth, to make known with boldness the mystery of the gospel for which I am an ambassador in chains, so that I may have the courage to speak as I must.
The first passage, in particular, has something of what would be called "motherhood and apple pie" if it were found at the end of a speech given by a university chaplain at a congregation ceremony.
But though St. Paul doesn't write boilerplate, and he certainly did mean the Thessalonians should pray without ceasing, that is clearly not the particular theme of this part of the letter, much less the letter as a whole. It would fall to subsequent generations of Christians to resolve the theoretical appeal of praying always with the practical difficulties.
Here let me just point out that prayer is not the only thing we are enjoined to do always in 1 Thes 5:17:
Rejoice always. Pray always. Give thanks always.
Thanksgiving is a form of prayer (remember ACTS: adoration, confession, thanksgiving, supplication), and it may be only through always giving thanks that one can always rejoice, which is not so much a form of prayer as a fruit of the Holy Spirit ripened through prayer.
But if we want to pray always, we'd better be prepared to rejoice and give thanks always, too.