Did the Messenger (blessings and peace of Allah be upon him) engage in recreational activities? Or was his entire life filled with acts of obedience and worship?

Praise
be to Allah.

Yes: the Prophet (blessings and peace of Allah be upon him) used to engage
in recreational activities, yet his entire life was devoted for Allah, the
Lord of the Worlds, as Allah, may He be glorified and exalted, says
(interpretation of the meaning): “Say (O Muhammad (blessings and peace
of Allah be upon him)): “Verily, my Salat (prayer), my sacrifice, my
living, and my dying are for Allah, the Lord of the Worlds” [al-An ‘aam
6:162]. 

His noble life was all for the sake of Allah (blessings and peace of Allah
be upon him). As for worship, it was obviously for the sake of Allah. 

With regard to matters other than worship, the Prophet (blessings and peace
of Allah be upon him) used to race with ‘Aa’ishah, and he used to joke with
his family, treat them kindly and chat with them at night. 

Muslim (670), Abu Dawood (1294) and an-Nasaa’i (1358) narrated from Jaabir
ibn Samurah that the Messenger of Allah (blessings and peace of Allah be
upon him) would not get up from the place in which he had prayed Fajr until
the sun had risen, and when the sun had risen he would get up. They used to
chat and talk about matters of the Jaahiliyah, and they would laugh but he
smiled.. 

At-Tirmidhi (1990) narrated – and classed the report as saheeh – that Abu
Hurayrah said: They said: O Messenger of Allah, you joke with us. He said:
“But I never say anything but what is true.”

Classed as saheeh by Shaykh al-Albaani in Saheeh al-Adab al-Mufrad, no.
265. 

Ibn Maajah (144) narrated from Ya‘la ibn Murrah that they had gone out with
the Prophet (blessings and peace of Allah be upon him) to a meal to which
they had been invited, and there was al-Husayn playing in the street. The
Prophet (blessings and peace of Allah be upon him) came in front of the
people and stretched out his hands, and the child started to run here and
there. The Prophet (blessings and peace of Allah be upon him) made him laugh
until he caught him, then he put one hand under his chin and the other on
his head and kissed him.

Classed as saheeh by al-Albaani in Saheeh Ibn Maajah. 

It
was narrated from Anas ibn Maalik that a man from among the desert people –
who was called Zaahir ibn Haraam – used to give gifts to the Prophet
(blessings and peace of Allah be upon him), and the Messenger of Allah
(blessings and peace of Allah be upon him) would equip him if he wanted to
go out (on campaign). The Messenger of Allah (blessings and peace of Allah
be upon him) said: “Zaahir is our man of the desert, and we are his people
of the city.” 

The Prophet (blessings and peace of Allah be upon him) came to him when he
was selling his merchandise, and embraced him from behind. The man did not
see him, and he said: Let me go! Who is this? Then he turned to face him and
when he realised that it was the Prophet (blessings and peace of Allah be
upon him), he placed his back against his chest. The Messenger of Allah
(blessings and peace of Allah be upon him) said: “Who will buy this slave?”
Zaahir said: O Messenger of Allah, you will find me unsellable. He said:
“But with Allah, you are not unsellable”; or he (blessings and peace of
Allah be upon him) said: “But with Allah you are valuable.” 

Narrated by Ahmad in al-Musnad (12468 – Risaalah edition); Ibn
Hibbaan (5790); at-Tirmidhi in ash-Shamaa’il (239); classed as saheeh
by al-Albaani. 

Ahmad (24334) narrated that ‘Aa’ishah said: The Messenger of Allah
(blessings and peace of Allah be upon him) said on that day – i.e., the day
when the Abyssinians played in the mosque: “The Jews should know that there
is room for leisure in our religion. I was sent with a tolerant,
monotheistic religion.”

Classed as saheeh by al-Albaani in Saheeh al-Jaami‘ (3219). 

And there are many similar reports. 

Imam at-Tirmidhi (may Allah have mercy on him) included two useful chapters
in his book Shamaa’il an-Nabi (sall Allahu ‘alayhi wa sallam), one of
which is: Chapter on what was narrated about the smile of the Messenger of
Allah (blessings and peace of Allah be upon him) and the other is: Chapter
on what was narrated about the way in which the Messenger of Allah
(blessings and peace of Allah be upon him) joked. 

A
person may sleep or play or engage in foreplay with his family, and if his
intention in doing so is good, then he will be rewarded for it. 

Al-Bukhaari (4345) narrated that Mu‘aadh ibn Jabal (may Allah be pleased
with him) said: As for me, I sleep and I pray qiyaam, and I seek reward for
my sleep just as I seek reward for my qiyaam. So how about the Messenger of
Allah (blessings and peace of Allah be upon him)? 

For more information, please see the answer to question no.
143026.