A typical plot in a Dom Com, when a middle-aged woman realizes that menopause is right around the corner and that she doesn't have much time left to have children if she hasn't already, or that she won't be able to have more. Given the Law of Inverse Fertility, the harder she tries to become pregnant the less likely she will be to actually get pregnant, but the moment she decides that she doesn't want a baby she will suddenly become pregnant.
Common plots involve a woman looking into sperm donation or adoption, or if she is approaching Christmas Cake status, attempting to get married before her 'expiration date.' As in Real Life, the latter is not a good idea and almost always ends disastrously.
And Heaven help the poor thing if she also has parents who are screaming "I Want Grandkids!"
An Older Than Feudalism variant is the woman who wishes for a child and proceeds to have a Wonder Child. See also Mandatory Motherhood.
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An entire story arc of Rosario + Vampire focuses on Mizore's problems because of this. Specifically, her race can only reproduce through her mid-twenties, and was nearly forced into an arranged marriage to preserve her race. Break the cutie ensues. But she gets better.
This is part of one of Lisa's monologues in My Cousin Vinny, with a beautiful counter from Vinny.
Inverted in 24 Hour Party People when Tony Wilson expresses great desire to be a father while talking with his first wife.
In A Brother's Price Eldest Whistler is twenty eight and has time left before menopause, plus her younger sisters have more, but she'd still like to marry and have one child, to see what pregnancy is like for herself. A teacher, Miss Skinner, feels this trope far more strongly, to the point of desperation; another character sighs and calls her addled, saying she should have gone to the cribs long ago. It's noted that not all women get this kind of feeling. Said other character can't stand children.
Live Action TV
Angel: Illyria is revealed in the Angel comic series to have a once-in-a-millennium mating cycle, signalled by Illyria going 'into heat'.
Monica was babycrazy practically from the start, and especially from the first season finale, and it led to the end of her relationship with Richard (played by Tom Selleck) since he didn't want kids. Later seasons saw her and Chandler adopt after they had trouble conceiving, though ironically Courtney Cox really was pregnant when the adoption episodes were filmed.
Phoebe didn't seem to have these feelings until she served as a surrogate for her brother (a plot necessitated by the actress' real life pregnancy), at which point she started wishing for her own family.
Rachel's feelings on the subject seemed to be limited to "be married and have a baby before I turn thirty-five", which she did get, in a roundabout way.
On Sabrina the Teenage Witch, aunt Hilda once had a problem with her biological clock (a literal magical clock) and began a series of desperate measures to try and become pregnant, including randomly choosing a guy off the street to marry. Sabrina eventually offered her own clock as a substitute for Hilda's, but luckily it all worked out in a timely fashion.
In an episode of Law & Order: SVU involving a stolen cryotank of embryos Olivia comes face to face with her own anxiety about having children and reveals that she had been turned down by adoption agencies for not having an extended family network. Her partner Elliot offered to help her in whatever way he could... whatever that means.
Also invoked when a deliberately pregnant teen (the result of a pregnancy pact with friends) taunted middle-aged, childless Olivia, convinced that her disapproval of the girl's choices was only jealousy. She rubbed her swollen belly and chanted mockingly in Olivia's face, "Tick-tock. Tick-tock."
Liz on 30 Rock in "The Baby Show". Tina Fey later made a whole movie about it called Baby Mama.
Liz, basically constantly, complete with her mom who wants grandkids. She goes through several boyfriends, considers adoption and sperm donation, buys a wedding dress while single....yeah. Two of those things happened in Season 1. (She finds a worthwhile boyfriend in Season 6.)
Even Jack isn't immune, with his mother pressuring him to get married (again) and have children (and then criticizing every woman he dates). He actually marries Avery Jessup in Season 4, and has a daughter in Season 5...only to spend all of Season 6 with Avery arrested in North Korea, leading to their divorce upon her return.
Though it wasn't necessarily due to age, one of the lesbian couples on The L Word decided they wanted a child in this sort of plot.
Actually happens to a man in the backstory of Power Rangers Operation Overdrive, in which Adventurer Archaeologist Andrew Hartford decides his clock is ticking, and so he builds himself a Ridiculously Human Robot teenage son (Mack, who, incidentally, becomes the team's Red Ranger). Being a childrens' show, why he doesn't use his ungodly amounts of money to get a gold-digger wife is never explored, but it at least gets a lampshade:
Andrew: I was too busy with my work to find the right woman.
Mack: Why didn't you make one of those, too!?
Naturally, that is not the most popular fan theory.
House had a sideplot involving Cuddy trying to get pregnant via sperm donors and fertility treatments. The title character handles this with his usual variance of sensitivity, ranging from agreeing to inject her in the backside with a hormone treatment to actually using her infertility as a vulnerable point to lash out at her. In the end, she adopts.
From Spin City, Carrie is so desperate to have a baby that she ends up stealing Mike's sperm.
Dr. Brennan in Bones had no interest in parenthood during the first few seasons, until towards the end of Season Four, when she decided she wanted to be a mother, and asked Booth to volunteer as a sperm donor. This plotline was interrupted by Booth's brain tumor, and made academic by the end of Season Six, when Booth and Bones conceived a child the old-fashioned way.
On Strong Medicine, Dr. Dana Stowe isn't bothered to much by still being single in her mid-to-late thirties, but she is bothered by still being childless—especially since as an OB/GYN and fertility specialist, she knows full well the risks and difficulties of conceiving and maintaining a pregnancy at an older age.
This is running theme for Marge's sister Selma in The Simpsons, who at various points had considered sperm donation, dated several men, had been married several times and took on an iguana as a substitute. She eventually adopted a baby girl named Ling from China.
Sealab 2021: "Chickmate" revolves around Debbie's clock going off. She wakes up screaming "I want a baby!", briefly treats a dolphin like her baby, then starts interviewing all of the males on Sealab to find a suitable father. She eventually gets turned off the idea by coming to the conclusion that the guys are childish enough.
In a very not funny example, Harley Quinn and the Joker jokingly reference this is Batman Beyond: Return of the Joker in a flashback, only to then laugh at the concept of "the joy of childbirth". The result is an attack against Batman that involves a Mind Rape against Tim Drake that turns him into a mini-Joker and leaves him mentally scarred for life. At the very end of the movie it's also revealed that Harley has two granddaughters named Delia and Deidre Dennis, meaning that at some point she did actually have at least one kid. It is unknown whether the "Dee Dees" are descended from the Joker as well.
An episode of Family Guy showed a 37-year old woman, desperate to get pregnant, on a date with an ex-convict.
In the series finale of Daria, after Mrs. Barch somewhat cornered Mr. O'Neil into proposing to her, she was very vocal about having a short engagement and less than subtle about the reasons why.
Mrs. Barch: We need to get cracking before my eggs dry out.
Truth In Television
This is so widely-recognized that some languages have a word for it. For instance, in German, the word is Torschlusspanik—literally "panic at the closing of the gate". It can be used in other situations where an opportunity is coming close to an end, but it's most commonly used for women's ticking biological clocks.